We recently had two nights in NYC with my niece who is studying musical theatre and she loved Hairspray and Avenue Q. We would have seen Mama Mia if there were tickets available - we have seen it and loved it. Avenue Q does have a few racy moments but no nudity or real oscenity - certainly nothing worse than they have seen or heard elsewhere.
Don't forget the TKTS booth, but use the one near South Streeet Seaport - address is listed as Front and Water but it is actually closer to Maiden Lane. Rarely a line, and if you are in th market for matinee tickets they have them the day prior. Remember that it is cash only.

I vote for Rent because it's such a great musical and teenagers LOVE it (literally they wait in line for tickets every day), but if you can't get tickets to that, then Wicked or A Chorus Line would be great. I wouldn't bother with Mamma Mia...not a very good musical although some people tend to love it! And Mamma Mia's demographic tends to be towards older women...not young girls.

Wicked, Chorus Line, Rent.

I saw Chorus Line for my 16th birthday, and LOVED it. Now I should go back & revisit it. Maybe I'll feel young again.. ;-)