What do you call someone who is being mentored? Is it mentoree or mentee? Does the term student or pupil imply a context outside the business environment?
I think you may be looking for protege:
protege — a person who receives support and protection from an influential patron who furthers the protege's career
Some businesses provide less experienced staff with mentors. I have heard the mentors refer to their "mentees". Wikipedia says this is a recent term.
There's nothing at all wrong with mentoree, as these 1270 written instances in Google Books show. The fact that most dictionaries don't explicitly list this particular inflection of mentor is of no consequence - it's used often enough already, and will always be understood even at first sight.
Mentee also occurs quite often in written form - but I have to say I don't like that version, and it seems to me it won't always be so readily understood. Bearing in mind that what a mentor does is mentoring, not menting, I think it's a badly-formed coinage anyway.
I would also point out that to mentor as a verb form barely existed before the last couple of decades, so it's hardly surprising more exotic inflections still encounter some resistance. But I've no doubt the erroneous mentee will die off, and "regular" mentoree will soon be firmly established.
EDIT: My Chambers dictionary 1983 has only the short entry "a wise counsellor" for mentor. The latest 2011 edition is much expanded, fully describing the "office mentor" role, and noting its use as a verb. I don't have the two intermediate editions to hand, to say when the verb use was first noted. The point is, we're looking at rapidly changing language here; dictionaries may not always keep up.
I also tip my hat to @horatio for pointing out that since the word comes from Mentor, counsellor to Homer's Odysseus (later, tutor to his son), arguably OP's answer is Odysseus (or Telemachus)
I know that mentee is common in American English, and is used by a variety of mentoring programs. One example of such is the US Department of Health and Human Services's Mentoring Program.
I know several people that work in this field, and they all use the word mentee.
A few other organizations that use "mentee":
I assume you are asking for a word for a person guided by a mentor. "Student" or "pupil" perhaps, but they have the implication of teaching. Perhaps "acolyte" or "disciple", though the latter implies devotion, and is used for follwers of a religous leader. Mentee is indeed "modern" and not widely used.
Mentee may be in the dictionary, but it is not in common usage (in my experience). My browser spell-check doesn't even recognize it :) It'll probably get the point across, though it may earn some odd looks.
Mentoring someone is not strictly a teaching relationship, so student and pupil don't quite fit. Especially in a business setting, a mentor is often just there to help the person get settled, show them particulars of your business and answer questions. This is not "teaching" them to do their job except in the loosest sense.
In short, I don't think there's a single nice term that mirrors trainee.