This site Grandparents.com has some great suggestions.
This question worries a lot of mentors because they imagine themselves stranded in a room with a young person who either just stares at them or can, at best, manage only a one- or two-word response to any attempt at conversation. This scenario does arise, but it is preventable—or at the least is one you can successfully negotiate, especially with help of Michael Karcher and those who have studied the “I dunno, what do you want to do?” syndrome (Karcher, 2010; Karcher, Herrera & Hansen, 2011). Just bear three things in mind.[faq category=78 template=accordion]
Many programs and case managers are there to suggest structured training and activities offered by your program coordinator. Most are good, and many are getting even better at ensuring that you get the training and ongoing support needed to address the “what to do” question.
If you listen, they will find a way to tell you what they want to do. Let their interests be your guide.
Many mentors worry that they will fail the kids they are mentoring or that their mentees will think they are “uncool.” On the latter count, harbor no doubt. All but the youngest mentees will not think you are cool (happening, hip, groovy, awesome, da bomb, sweet, or whatever word applies in your own generational lingo). “Cool,” however, is not what most young people seek. They are looking for adults who have a genuine interest in them and are competent enough in the ways of the world to help them get where they want to go.