Maybe... maybe not... many of the games and other activities are geared to a little older audience. However, here's a seemingly independent source of age suitability games for Xbox 360...
https://www.commonsensemedia.org/game-reviews
Additionally, you may wish to consider the Xbox One, since it's designed to supplant the Xbox 360 to the point that Xbox 360 games won't work on the Xbox One... Here's a realistic discussion on that point:
http://support.xbox.com/en-US/xbox-one/system/compatibility-with-xbox-360
Lastly, the Xbox One (new this year) often stores the games rather than plying them from any media, which means you may have to buy additional memory as time goes on, but that's readily available already and reasonably priced...
Good luck!