I think you should hitup the bargain basement! Also you should get a HDD if you haven't already to install your games if you prefer a quieter experience.
Condemned 1/2 (Truely unique, the first more than the 2nd are first person psychological horror brawlers)
Kameo! You can play it two player and its a really fun experience. A bit of a gem and a really low price! Its made by Rare, not too violent and can be enjoyed with others.
Crackdown -> This is truely an action sandbox game, not perfect but its bargain basement prices and fun atmosphere are really something unique.
Blue Dragon -> Is quite fun to play, Lost Odyssey is more grown up but your familiy may enjoy the child protaginists of Blue Dragon.
Ninja Gaiden 2 -> My pick compared to Devil May cry because of its smooth as silk controls and fair but difficult combat. Its easier because you don't have to learn wacky button layouts unlike DMC4. Its also bargain bin pricing! 
Dark Sector -> Its not brilliant but it gives you a unique gameplay angle with the bladed frisbee thing. It was a lot of fun for at least one playthrough, its kinda a Gears clone with a twist.
You should check out some online games if possible: Worms HD (mega fun), Brain (Incredibly unique platformer) and Caste Crashers (Fun for the whole family)
Oh one last thing if your familiy loves Quiz games then IMO its definately worth it to get that online enabled and grab a Live subscription as 100 vs 1 I hear is pretty fantastic. Oh yeah further on the online thing, if you have netflix you can stream movies instantly to your Xbox 360 as well with a gold subscription.