I would stay far away from Portal 2. While it's a good game, I'm going to assume two things. One, you have probably played it already. Two, she probably doesn't have much experience with 3D puzzles or timing puzzles either. I imagine it will be very frustrating for both of you, either because you are bored because she hasn't gotten the puzzle yet or she gets bored because you just tell her what to do. Stay far far away.
Also, for me, being a girlfriend myself, it's not so much the game, but how you two interact while playing it. Pick something you love and know well. She's probably interested in getting into gaming because you're into gaming; she's trying to find something y'all can share. How much of a beginner is she? If she is an absolute beginner, doesn't even play Farmville, type of girl, then a party game would be the best bet. They're meant for anyone to pick up and have fun, and it will establish a good start for you two. WarioWare is fun, and a bit more "game-y" than Just Dance. (At least it's part of a major franchise, right?) If, however, she is like most girls today who has some experience with games, then I don't see why you can't dive into Halo or CoD or something. Start off in either single player or split-screen just to let her grasp the mechanics. Once she has a good grasp of what buttons do what, give her some basic tips, then head into online play. Be sure to turn off the chat, and encourage her even for small victories. Let her know that it's ok to die 50 bajillion times. Don't condescend, but congratulate her when she gets her first assist or kill or whatever. Things that seem insignificant to you because you've been playing games so long will be a big deal to her. So long as you remain calm and encouraging, even if she sucks total balls the first time, she will be likely to come back and play again.
