How to call a function in response to a Button's OnSelect event?

Edit: I added the OnSelect version.

Note that selecting and highlighting / hovering are two different things. When you hover over your button with mouse, OnPointerEnter will be triggered (if using Color Tint transition, Highlighted color will be also activated), and when you exit the Button, OnPointerExit will be triggered. But when you click a Button or navigate to it with keyboard, it is selected and when you click elsewhere or navigate away from your Button, it gets deselected.

So I think you might have to think about your highlighting / hovering if you want to make it work for different navigation styles. When using mouse one doesn't expect he/she has to click a button to see a tooltip.

To detect Button select with ISelectHandler (this should work for at least keyboard and mouse):

public class ButtonTest : MonoBehaviour, ISelectHandler
{
    public void OnSelect(BaseEventData eventData)
    {
        Debug.Log("selected");
    }
}

And seems like you also might need deselect, if that is the case, then you can also implement IDeselectHandler:

public class ButtonTest : MonoBehaviour, ISelectHandler, IDeselectHandler
{
    public void OnSelect(BaseEventData eventData)
    {
        Debug.Log("selected");
    }
    
    public void OnDeselect(BaseEventData eventData)
    {
        Debug.Log("deselected");
    }
}

If you want to detect cursor hovering over your Button use IPointerEnterHandler / IPointerExitHandler:

public class ButtonTest : MonoBehaviour, IPointerEnterHandler, IPointerExitHandler
{
    public void OnPointerEnter(PointerEventData eventData)
    {
        Debug.Log("pointer enter");
    }

    public void OnPointerExit(PointerEventData eventData)
    {
        Debug.Log("pointer exit");
    }
}

You will also have to add a using statement for UnityEngine.EventSystems for these to work.

If you want to detect Button select without using ISelectHandler and IDeselectHandler, you could also do this in your Update:

void Update()
{
    if (EventSystem.current.currentSelectedGameObject == this.gameObject)
    {
        Debug.Log("selected");
    }
}