Since there is a 10 yr old in my household I would recommend the following games if they are interested in these types of:
Music / Dance: Get Xbox Kinect to play Dance Central - No Controllers required!. Continuing with music Rock Band set along with the game should suffice.
RPG: Halo Reach is good, not too violent. Stay away from Black Ops and any other Call of Duty games.
Car Racing games: Burnout Paradise is a great game which gives the player incentive to keep playing. Appropriate for all ages.
Lego games: Star Wars (if interested in star wars), Indiana Jones
Sports games: Madden 2011, NBA 2k11, NBA Live 2011 are all excellent games for xbox 360 depending on which sport is preferred by the child.
Skateboarding: If they are interested in skateboarding then try Tony Hawk's Ride. It is a little hard but it gets easier as you play it more often.