Who'd be a mentee?

This much is well-known. Mentor comes from the story of Odysseus. While he’s finding his his way back from Troy, his wife, Penelope, is struggling make ends meet in Ithaca, fending off the suitors who are trying to convince her that Odysseus is dead and she needs a new husband.  His son, Telemachus, is a teenager at the time, and feeling he ought to be the man of the house, but not very sure of himself. So the gods, who are watching over all this, decide to give Telemachus some help, and they offer guidance by speaking through one of Odysseus’ old-time trusted friends. And (you’re way ahead of me) that friend’s name was Mentor.

If you're taking advice from a mentor, you should be a telemachus (pl. telemachoi maybe), right? You can see why people looked for a shorter alternative. They (presumably) thought mentor fit the agent-object pattern that you see in word pairs like employer-employee. That pattern came from French into English, where the -er (orig. -eur) ending is the agent, the employer, and the -ee (orig. -é or -ée) participle ending signifies the object: employee, the person employed. Applying the same French-derived agent-object pattern (sort of) gives you mentee.

Except, of course, it's not a mentor that's the agent of a mentee, but a menteur.

If you're a mentee, while you're thinking a better word for yourself, take the advice of your mentor critically. Are you hearing the voices of the gods? Or something else?

And as I write this I wonder if menter derives from the same root, and the French, in their Voltairian wisdom, treated the words of the Gods as untrue right from the get-go.

Probably not, but the golden rule applies. Trust, and verify. Mentors are in the end only human.