I would highly recommend that you not do that. The Bible says that we are to make disciples of all nations, baptizing them in the name of the Father, the Son, and the Holy Ghost. Jesus never used the term mentorship, but what we know for sure is that he meant for us to make our own protégés. He wanted us to make followers who would learn the doctrines and then teach them to their own followers. For a more detailed discussion on the difference between mentoring and discipleship, see Chapter 2.
Here's the deal: Jesus wanted his followers to make more followers. But before they could be followers of Peter, Matthew, James, John, and the others, they had to be followers of Christ. Before a young believer can be your protégé, he must be God's son by adoption. Then he is ready to become a protégé so that he can be a good disciple.
The reason you don't want to enter a mentoring relationship with an unbeliever in the hopes that he will convert to Christianity is that it probably won't happen that way. If you attempt to bring Jesus into the discussion every time something comes up, then you'll find yourself undoubtedly turning the would-be believer off to the Gospel. It might even make the young man think that you're only mentoring him so that you can try to convert him. That’s not fair to anyone in the picture, including yourself and Christ. Your goal is to make followers of Christ, but you shouldn’t use a mentoring relationship to do so. Win them to Christ first, and then mentor them on what it means to be a Christian.
You must understand, however, that I do mentor unbelievers. As a Chief Petty Officer in the Navy, I am constantly guiding my staff in what I hope will be a successful career. I am quite adept at understanding the difference between a Christian mentoring relationship and one of a secular nature. I can certainly guide a Sailor in making competent career choices, and even competent personal choices, and remain on a neutral field. I would certainly hope someday to be able to share the Gospel, but only as I would hope to one day share the Gospel with all of my unbelieving friends, not just the ones I mentor. The important thing to remember is that while it is certainly ok to hope to share the Gospel with a protégé, it shouldn’t be the reason you enter into a mentoring relationship.