The New York Times seems to have decided that "mentee" is a word, for today's paper applies the term to Paula Broadwell (whose sometime mentor was General David Petraeus), and the other week the Times applied it to one of the unfortunate women who imagined that Bill Cosby was her mentor. I dislike the expression, as implying that there is verb stem out there something like "ment." Yet "mentir" and "mentior" don't seem to be what anyone has in mind.
Skeat,, however, after giving the Homeric character Mentor as the source, relates the word to "monitor", derived from "monere", to warn. Well, he was a scholar, and I am not. Yet the nearest I can find in Liddell and Scott is μηνυτήρ, "informer, guide", related to the verb μηνύω, "disclose what is secret, reveal". Most of the cognate words seem to have to do not with guiding someone, but with giving incriminating information to the authorities.