The "root" of the word mentor is a character in the Odyssey, not the verb "to ment", so it's nonsense to talk of a "mentee". A mentor might have a pupil, or perhaps a protégé, but not a mentee.
Edit: I've looked it up and it seems the most common terms for someone who has a mentor are "apprentice" or "disciple". We need to disabuse ourselves of the notion that such relationships need to be described with words that share a root.
d.