My girlfriend doesn't really play games on her own, but she did have an NES growing up, and recently has played games like Limbo and Psychonauts. The games we enjoy playing together most are Castle Crashers and Splosion Man. Splosion Man might be difficult for someone who hasn't really played games before, because it relies heavily on timing and may be frustrating for someone who isn't experienced with the controls.
Castle Crashers on the other hand, is very easy to pick up and play. Like most brawlers, the gameplay is pretty repetitive, but still fun, which let's you converse while playing. That seems to be an important feature of a good girlfriend game. The cute artwork doesn't hurt either. Both of the above games have trials, so there's no risk in giving them a shot.
We played the Wallace & Gromit games as well as some Sam & Max on the PC, but they're also available on the 360. They're not multiplayer, but we had fun passing the controller and figuring out the puzzles together.
We also play quite a bit of Scene It and Rock Band, but I'm not sure if those are the kind of games you're looking for.
edit: I was just looking through my played games list and remembered we rented The Simpsons Game and she really liked it. I thought the gameplay was so-so, but it was fun to play with her and it was pretty funny. We definitely had a good time. It's probably pretty cheap now and I believe there is a demo.