How do I change the face in the people page of Photos?

When you change the face, as recommended by Mark, pick one of the faces that have been detected automatically by Photos, not a face you had to add manually. Then your selection will be more stable after a system upgrade or when you are syncing your Photos Library with iCloud Photos across devices. Manually added faces may occasionally vanish. Pick a photo as the key photo, where the face is large enough for the huge thumbnails and distinctive and you can easily recognise the face (for example, if you have albums of siblings that are looking similar).

Open up the page for that person, right-click (option-click) on the picture you want to use, and select "Make Key Photo" from the context menu.