The word sounds a bit silly to my ear, but the younger person can be called a mentee, the counterpart to a mentor.
a person who is advised, trained, or counselled by a mentor. (Oxford Dictionaries)
Over the course of five years this evolved to a more casual adolescent to adult-model relationship in which the preservice teacher helped their mentee set goals, shared college experiences, and chaperoned college campus tours.
-Radcliffe, Richard and Bos, Beth, "Mentoring approaches to create a college-going culture for at-risk secondary level students." American Secondary Education. 2011, Vol. 39 Issue 3, p86-107.
In our context the older person might be called a youth mentor, to emphasize this particular sense, rather than the sense that entails professional training. But I've never heard an analogue for mentee that eliminates the corresponding ambiguity. (One can find about 1000 hits for youth mentee on Google, but this sounds awfully stilted to me, and I'd anyway avoid the term because it's potentially confusing enough to demand more explanation.)
The organization Big Brothers Big Sisters (the fraternal metaphor is explicit here), which arranges this sort of mentorship between adult volunteers and children, uses the peculiar term Littles, but I've never heard this usage elsewhere, and again without more context it would surely cause confusion.
International research has shown that positive relationships between Littles and their Bigs have a direct and measurable impact on children's lives and of the community as a whole.