My current code is as follows:

/*
 * Buttons Menu Sketch
 * Three buttons with a 1602 i2c LCD Display
 * 2 buttons for up/down and one button for select
 * Variation of the code on https://www.instructables.com/id/Arduino-LCD-16x2-Turn-Single-Screen-Into-Multiple-/
 * copy right statement goes here
 */

 
#include <LiquidCrystal_I2C.h>
#include <Wire.h>

#include <EEPROM.h>     //EEPROM Functions: EEPROM Module
#include <Arduino.h>    //EEPROM Functions: for type definitions

LiquidCrystal_I2C lcd(0x27,16,2);


//Globals:
int page_counter  = 1 ; //To move between pages
int page_counter_s  = 1;  //submenu

int up = 2;       //Up button
int down = 3;     //Down button
int selection = 4;    //Select button

int LED = 13;

int pageCount   =   7;  //Main Menu

 
/*
Storage debounce function
*/
 
//Main Menu 
boolean last_up     =   LOW;            
boolean last_down     =   LOW;
boolean last_select   =   LOW;

boolean current_up    =   LOW;  
boolean current_down  =   LOW;
boolean current_select  =   LOW;

boolean last_up_s     =   LOW;            
boolean last_down_s     =   LOW;

boolean current_up_s    =   LOW;  
boolean current_down_s  =   LOW;



//---- De-bouncing function for all buttons----//
boolean debounce(boolean last, int pin){
  boolean current = digitalRead(pin);
  if (last != current){
    delay(5);
    current = digitalRead(pin);
  }
  return current;
}

//Menu Function
void Menus(){                                                                                                                                     

  current_up = debounce(last_up, up);         //Debounce for Up button
  current_down = debounce(last_down, down);   //Debounce for Down button
  current_select = debounce(last_select, selection); //Debounce for Select button
  //----Page counter function to move pages----//
  
    //Page Up
    if (last_up == LOW && current_up == HIGH){  //When up button is pressed
      lcd.clear();                     //When page is changed, lcd clear to print new page      
      
      if(page_counter < pageCount){             //Page counter never higher than 3(total of pages)
        page_counter= page_counter +1;   //Page up
      }
      else{
        page_counter = pageCount;  
      }
    }
    
    last_up = current_up;

    //Page Down
    if (last_down== LOW && current_down == HIGH){ //When down button is pressed
      lcd.clear();                     //When page is changed, lcd clear to print new page    
      if(page_counter >1){              //Page counter never lower than 1 (total of pages)
      page_counter= page_counter -1;   //Page down
      
      }
      else{
      page_counter= 1;  
     }
    }
    
    last_down = current_down;

  


//------- Switch function to write and show what you want---// 
  switch (page_counter) {

   
    case 1:{ //Each case is a menu option
      lcd.setCursor(0,0);
      lcd.print("Counter 01:");
      lcd.setCursor(0,1);
      lcd.print("100 Hrs");
    }
    break;

    case 2:{ //Each case is a menu option
      lcd.setCursor(0,0);
      lcd.print("Counter 02:");
      lcd.setCursor(0,1);
      lcd.print("70 Hrs");
    }
    break;
  
  case 3:{ //Each case is a menu option
      lcd.setCursor(0,0);
      lcd.print("Reset C01");
      lcd.setCursor(0,1);
      lcd.print("Timer");
    
    //Action:
    if (last_select == LOW && current_select == HIGH){ //When down button is pressed
      lcd.clear();
      lcd.setCursor(0,0);
      lcd.print("C01 Timer reset");
      lcd.setCursor(0,1);
      lcd.print("was successful");
      
      digitalWrite(LED,HIGH);
      delay(500);
      digitalWrite(LED,LOW);
      delay(3000);
      lcd.clear();
    }
    }
    break;
  
  case 4:{ //Each case is a menu option
      lcd.setCursor(0,0);
      lcd.print("Reset C02");
      lcd.setCursor(0,1);
      lcd.print("Timer");
    
    //Action:
    if (last_select == LOW && current_select == HIGH){ //When down button is pressed
      lcd.clear();
      lcd.setCursor(0,0);
      lcd.print("C02 Timer reset");
      lcd.setCursor(0,1);
      lcd.print("was successful");
      
      digitalWrite(LED,HIGH);
      delay(500);
      digitalWrite(LED,LOW);
      delay(3000);
      lcd.clear();

    }
    }
    break;
  
  
  case 5:{ //Each case is a menu option
      lcd.setCursor(0,0);
      lcd.print("SET C01");
      lcd.setCursor(0,1);
      lcd.print("Interval");
    //Action:
    if (last_select == LOW && current_select == HIGH){ //When down button is pressed
      
       
      submenu();
    }
  }
    break;
  
  case 6:{ //Each case is a menu option
      lcd.setCursor(0,0);
      lcd.print("SET C02");
      lcd.setCursor(0,1);
      lcd.print("Interval");
    
    //Action:
    if (last_select == LOW && current_select == HIGH){ //When down button is pressed
      
         submenu();
        
    }
    }
    break;
  
  case 7:{ //Each case is a menu option
      lcd.setCursor(0,0);
      lcd.print("Exit Service");
      lcd.setCursor(0,1);
      lcd.print("Mode");
    
    //Action:
    if (last_select == LOW && current_select == HIGH){ //When down button is pressed
      
      lcd.clear();
      lcd.setCursor(0,0);
      lcd.print("Exiting");
      lcd.setCursor(0,1);
      lcd.print("Please Wait");
      
       digitalWrite(LED,HIGH);
      delay(500);
      digitalWrite(LED,LOW);
      delay(500);
      
      delay(3000);
      lcd.clear();
      }
    }
    break;
    
  }//switch end
 
}

void submenu(){
  
  current_up_s = debounce(last_up, up);         //Debounce for Up button
  current_down_s = debounce(last_down, down);   //Debounce for Down button
  current_select = debounce(last_select, selection); //Debounce for Select button
  //----Page counter function to move pages----//
  
    //Page Up
    if (last_up == LOW && current_up_s == HIGH){  //When up button is pressed
      lcd.clear();                     //When page is changed, lcd clear to print new page      
      
      if(page_counter_s < 8){             //Page counter never higher than 3(total of pages)
        page_counter_s = page_counter_s +1;   //Page up
      }
      else{
        page_counter_s = 8;  
      }
    }
    
    last_up = current_up_s;

    //Page Down
    if (last_down == LOW && current_down_s == HIGH){ //When down button is pressed
      lcd.clear();                     //When page is changed, lcd clear to print new page    
      if(page_counter >1){              //Page counter never lower than 1 (total of pages)
      page_counter_s = page_counter_s -1;   //Page down
      
      }
      else{
      page_counter_s = 1;  
     }
    }
    
    last_down = current_down_s;

  


//------- Switch function to write and show what you want---// 
  switch (page_counter_s) {

   
  
  case 1:{ //Each case is a menu option
      lcd.setCursor(0,0);
      lcd.print("Set dealy To:");
      lcd.setCursor(0,1);
      lcd.print("100 hrs");
    
    //Action:
    if (last_select == LOW && current_select == HIGH){ //When down button is pressed
      lcd.clear();
      lcd.setCursor(0,0);
      lcd.print("New Dealy:");
      lcd.setCursor(0,1);
      lcd.print("100 Hrs.");
      
      digitalWrite(LED,HIGH);
      delay(500);
      digitalWrite(LED,LOW);
      delay(3000);
      lcd.clear();
    }
    }
    break;
  
  case 2:{ //Each case is a menu option
      lcd.setCursor(0,0);
      lcd.print("Set dealy To:");
      lcd.setCursor(0,1);
      lcd.print("200 hrs");
    
    //Action:
    if (last_select == LOW && current_select == HIGH){ //When down button is pressed
      lcd.clear();
      lcd.setCursor(0,0);
      lcd.print("New Dealy:");
      lcd.setCursor(0,1);
      lcd.print("200 Hrs.");
      
      digitalWrite(LED,HIGH);
      delay(500);
      digitalWrite(LED,LOW);
      delay(3000);
      lcd.clear();
    }
    }
    break;
  
  case 3:{ //Each case is a menu option
      lcd.setCursor(0,0);
      lcd.print("Set dealy To:");
      lcd.setCursor(0,1);
      lcd.print("300 hrs");
    
    //Action:
    if (last_select == LOW && current_select == HIGH){ //When down button is pressed
      lcd.clear();
      lcd.setCursor(0,0);
      lcd.print("New Dealy:");
      lcd.setCursor(0,1);
      lcd.print("300 Hrs.");
      
      digitalWrite(LED,HIGH);
      delay(500);
      digitalWrite(LED,LOW);
      delay(3000);
      lcd.clear();
    }
    }
    break;
  
  case 4:{ //Each case is a menu option
      lcd.setCursor(0,0);
      lcd.print("Set dealy To:");
      lcd.setCursor(0,1);
      lcd.print("400 hrs");
    
    //Action:
    if (last_select == LOW && current_select == HIGH){ //When down button is pressed
      lcd.clear();
      lcd.setCursor(0,0);
      lcd.print("New Dealy:");
      lcd.setCursor(0,1);
      lcd.print("400 Hrs.");
      
      digitalWrite(LED,HIGH);
      delay(500);
      digitalWrite(LED,LOW);
      delay(3000);
      lcd.clear();
    }
    }
    break;
  
  case 5:{ //Each case is a menu option
      lcd.setCursor(0,0);
      lcd.print("Set dealy To:");
      lcd.setCursor(0,1);
      lcd.print("500 hrs");
    
    //Action:
    if (last_select == LOW && current_select == HIGH){ //When down button is pressed
      lcd.clear();
      lcd.setCursor(0,0);
      lcd.print("New Dealy:");
      lcd.setCursor(0,1);
      lcd.print("500 Hrs.");
      
      digitalWrite(LED,HIGH);
      delay(500);
      digitalWrite(LED,LOW);
      delay(3000);
      lcd.clear();
    }
    }
    break;
  
  case 6:{ //Each case is a menu option
      lcd.setCursor(0,0);
      lcd.print("Set dealy To:");
      lcd.setCursor(0,1);
      lcd.print("600 hrs");
    
    //Action:
    if (last_select == LOW && current_select == HIGH){ //When down button is pressed
      lcd.clear();
      lcd.setCursor(0,0);
      lcd.print("New Dealy:");
      lcd.setCursor(0,1);
      lcd.print("600 Hrs.");
      
      digitalWrite(LED,HIGH);
      delay(500);
      digitalWrite(LED,LOW);
      delay(3000);
      lcd.clear();
    }
    }
    break;
  
  case 7:{ //Each case is a menu option
      lcd.setCursor(0,0);
      lcd.print("Set dealy To:");
      lcd.setCursor(0,1);
      lcd.print("700 hrs");
    
    //Action:
    if (last_select == LOW && current_select == HIGH){ //When down button is pressed
      lcd.clear();
      lcd.setCursor(0,0);
      lcd.print("New Dealy:");
      lcd.setCursor(0,1);
      lcd.print("700 Hrs.");
      
      digitalWrite(LED,HIGH);
      delay(500);
      digitalWrite(LED,LOW);
      delay(3000);
      lcd.clear();
    }
    }
    break;
  
  case 8:{ //Each case is a menu option
      lcd.setCursor(0,0);
      lcd.print("Back");
      lcd.setCursor(0,1);
      lcd.print("200 hrs");
    
    //Action:
    if (last_select == LOW && current_select == HIGH){ //When down button is pressed
      
      //Exit function code
    }
    }
    break;
    
  }//switch end
  
}




void setup() {
  // put your setup code here, to run once:
  lcd.init();
  lcd.backlight();
  lcd.setCursor(0,0);
  lcd.print("Menu Test");
  delay(3000);
  lcd.clear();  
}

void loop() {
  

  Menus();
}

My current code is as follows: [Note: Updated with a function to handle screen updates based on @jsotola]

#include <LiquidCrystal_I2C.h>
#include <Wire.h>

#include <EEPROM.h>     //EEPROM Functions: EEPROM Module
#include <Arduino.h>    //EEPROM Functions: for type definitions

LiquidCrystal_I2C lcd(0x27,16,2);


//Globals:
int page_counter    =   1 ; //To move between pages
int page_counter_s  =   1;  //submenu

int up = 2;             //Up button
int down = 3;           //Down button
int selection = 4;      //Select button
int LED =   13;

int pageCount   =   7;  //Main Menu

 
//Main Menu 
boolean last_up         =   LOW;            
boolean last_down       =   LOW;
boolean last_select     =   LOW;

boolean current_up      =   LOW;  
boolean current_down    =   LOW;
boolean current_select  =   LOW;

boolean last_up_s     =   LOW;            
boolean last_down_s     =   LOW;

boolean current_up_s    =   LOW;  
boolean current_down_s  =   LOW;

String line1;
String line2;


//---- De-bouncing function for all buttons----//
boolean debounce(boolean last, int pin){
  boolean current = digitalRead(pin);
  if (last != current){
    delay(5);
    current = digitalRead(pin);
  }
  return current;
}

//LCD Function

void printlcd(){
    lcd.setCursor(0,0);
    lcd.print(line1);
    lcd.setCursor(0,1);
    lcd.print(line2);
}

//Menu Function
void Menus(){                                                                                                                                                                                                                                                                           

    current_up = debounce(last_up, up);         //Debounce for Up button
    current_down = debounce(last_down, down);   //Debounce for Down button
    current_select = debounce(last_select, selection); //Debounce for Select button
    //----Page counter function to move pages----//
    
        //Page Up
        if (last_up == LOW && current_up == HIGH){  //When up button is pressed
          lcd.clear();                     //When page is changed, lcd clear to print new page        
          
            if(page_counter < pageCount){             //Page counter never higher than 3(total of pages)
                page_counter= page_counter +1;   //Page up
            }
            else{
                page_counter = pageCount;  
            }
        }
      
        last_up = current_up;

        //Page Down
        if (last_down== LOW && current_down == HIGH){ //When down button is pressed
          lcd.clear();                     //When page is changed, lcd clear to print new page    
          if(page_counter >1){              //Page counter never lower than 1 (total of pages)
          page_counter= page_counter -1;   //Page down
          
          }
          else{
          page_counter= 1;  
         }
        }
        
        last_down = current_down;

    


//------- Switch function to write and show what you want---// 
  switch (page_counter) {

   
    case 1:{ //Each case is a menu option
  
      line1 = "Counter 01:";
      line2 = "100 Hrs";
      printlcd();
    }
    break;

    case 2:{ //Each case is a menu option
      line1 = "Counter 02:";
      line2 = "70 Hrs";
      printlcd();
    }
    break;
    
    case 3:{ //Each case is a menu option
      line1 = "Reset 01:";
      line2 = "";
      printlcd();
      
      //Action:
      if (last_select == LOW && current_select == HIGH){ //When down button is pressed
            lcd.clear();
            line1 = "Success!";
            line2 = "C01 Reset";
            printlcd();
            
            digitalWrite(LED,HIGH);
            delay(500);
            digitalWrite(LED,LOW);
            
            delay(3000);
            lcd.clear(); 

        }
    }
    break;
    
    case 4:{ //Each case is a menu option
      line1 = "Reset 02:";
      line2 = "";
      printlcd();
      
      //Action:
      if (last_select == LOW && current_select == HIGH){ //When down button is pressed
            lcd.clear();
            line1 = "Success!";
            line2 = "C02 Reset";
            printlcd();
            
            digitalWrite(LED,HIGH);
            delay(500);
            digitalWrite(LED,LOW);
            
            delay(3000);
            lcd.clear();

        }
    }
    break;
    
    
    case 5:{ //Each case is a menu option
     
        line1 = "Set 01";
        line2 = "Interval";
        printlcd(); 

      //Action:
        if (last_select == LOW && current_select == HIGH){ //When down button is pressed
            lcd.clear();
            submenu();
        }
    }
    break;
    
    case 6:{ //Each case is a menu option
        line1 = "Set 02";
        line2 = "Interval";
        printlcd();

      //Action:
        if (last_select == LOW && current_select == HIGH){ //When down button is pressed
            lcd.clear();
            submenu();
        }
    }
    
    case 7:{ //Each case is a menu option
        line1 = "Exit service";
        line2 = "mode";
        printlcd();
        
      //Action:
        if (last_select == LOW && current_select == HIGH){ //When down button is pressed
            
            lcd.clear();
            line1 = "Exiting";
            line2 = "Please Wait";
            printlcd();
            
            digitalWrite(LED,HIGH);
            delay(500);
            digitalWrite(LED,LOW);
            delay(500);
            
            delay(3000);
            lcd.clear();
            }
    }
    break;
    
  }//switch end
 
}

void submenu(){
    
    current_up_s = debounce(last_up, up);         //Debounce for Up button
    current_down_s = debounce(last_down, down);   //Debounce for Down button
    current_select = debounce(last_select, selection); //Debounce for Select button
    //----Page counter function to move pages----//
    
        //Page Up
        if (last_up == LOW && current_up_s == HIGH){  //When up button is pressed
          lcd.clear();                     //When page is changed, lcd clear to print new page        
          
            if(page_counter_s < 8){             //Page counter never higher than 3(total of pages)
                page_counter_s = page_counter_s +1;   //Page up
            }
            else{
                page_counter_s = 8;  
            }
        }
      
        last_up = current_up_s;

        //Page Down
        if (last_down == LOW && current_down_s == HIGH){ //When down button is pressed
          lcd.clear();                     //When page is changed, lcd clear to print new page    
          if(page_counter >1){              //Page counter never lower than 1 (total of pages)
          page_counter_s = page_counter_s -1;   //Page down
          
          }
          else{
          page_counter_s = 1;  
         }
        }
        
        last_down = current_down_s;

    


//------- Switch function to write and show what you want---// 
  switch (page_counter_s) {

   
    
    case 1:{ //Each case is a menu option
      
      line1 =   "Set dealy to:";
      line2 =   "100 hrs";
      printlcd();
      //Action:
      if (last_select == LOW && current_select == HIGH){ //When down button is pressed
            lcd.clear();
            line1 = "New Delay:";;
            line2 = "100 hrs";
            printlcd();
            
            digitalWrite(LED,HIGH);
            delay(500);
            digitalWrite(LED,LOW);
            delay(3000);
            lcd.clear();
        }
    }
    break;
    
    case 2:{ //Each case is a menu option
      
      line1 =   "Set dealy to:";
      line2 =   "200 hrs";
      printlcd();
      //Action:
      if (last_select == LOW && current_select == HIGH){ //When down button is pressed
            lcd.clear();
            line1 = "New Delay:";
            line2 = "200 hrs";
            printlcd();
            
            digitalWrite(LED,HIGH);
            delay(500);
            digitalWrite(LED,LOW);
            delay(3000);
            lcd.clear();
        }
    }
    break;
    
    case 3:{ //Each case is a menu option
      
      line1 =   "Set dealy to:";
      line2 =   "300 hrs";
      printlcd();
      //Action:
      if (last_select == LOW && current_select == HIGH){ //When down button is pressed
            lcd.clear();
            line1 = "New Delay:";
            line2 = "300 hrs";
            printlcd();
            
            digitalWrite(LED,HIGH);
            delay(500);
            digitalWrite(LED,LOW);
            delay(3000);
            lcd.clear();
        }
    }
    break;
    
    case 4:{ //Each case is a menu option
      
      line1 =   "Set dealy to:";
      line2 =   "400 hrs";
      printlcd();
      //Action:
      if (last_select == LOW && current_select == HIGH){ //When down button is pressed
            lcd.clear();
            line1 = "New Delay:";
            line2 = "400 hrs";
            printlcd();
            
            digitalWrite(LED,HIGH);
            delay(500);
            digitalWrite(LED,LOW);
            delay(3000);
            lcd.clear();
        }
    }
    break;
    
    case 5:{ //Each case is a menu option
      
      line1 =   "Set dealy to:";
      line2 =   "500 hrs";
      printlcd();
      //Action:
      if (last_select == LOW && current_select == HIGH){ //When down button is pressed
            lcd.clear();
            line1 = "New Delay:";
            line2 = "500 hrs";
            printlcd();
            
            digitalWrite(LED,HIGH);
            delay(500);
            digitalWrite(LED,LOW);
            delay(3000);
            lcd.clear();
        }
    }
    break;
    
    case 6:{ //Each case is a menu option
      
      line1 =   "Set dealy to:";
      line2 =   "600 hrs";
      printlcd();
      //Action:
      if (last_select == LOW && current_select == HIGH){ //When down button is pressed
            lcd.clear();
            line1 = "New Delay:";
            line2 = "600 hrs";
            printlcd();
            
            digitalWrite(LED,HIGH);
            delay(500);
            digitalWrite(LED,LOW);
            delay(3000);
            lcd.clear();
        }
    }
    break;
    
    case 7:{ //Each case is a menu option
      
      line1 =   "Set dealy to:";
      line2 =   "700 hrs";
      printlcd();
      //Action:
      if (last_select == LOW && current_select == HIGH){ //When down button is pressed
            lcd.clear();
            line1 = "New Delay:";
            line2 = "700 hrs";
            printlcd();
            
            digitalWrite(LED,HIGH);
            delay(500);
            digitalWrite(LED,LOW);
            delay(3000);
            lcd.clear();
        }
    }
    break;
    
    case 8:{ //Each case is a menu option
      
      line1 =   "Back";
      line2 =   "";
      printlcd();
      //Action:
      if (last_select == LOW && current_select == HIGH){ //When down button is pressed
            lcd.clear();
            line1 = "Returning";
            line2 = "";
            printlcd();
            
            digitalWrite(LED,HIGH);
            delay(500);
            digitalWrite(LED,LOW);
            delay(3000);
            lcd.clear();
        }
    }
    break;
    
  }//switch end
    
}




void setup() {
  // put your setup code here, to run once:
  lcd.init();
  lcd.backlight();
  lcd.setCursor(0,0);
  lcd.print("Menu Test");
  delay(3000);
  lcd.clear();  
}

void loop() {
  

  Menus();
}