I got my girlfriend into games via Street Fighter 4 and Mass Effect. Might not work out the same for you.
Might be easier to do a tower defense type of game like Orcs Must Die 2 or Dungeon Defenders. Plants vs Zombies was a damn good one to pick.
My previous girlfriend got into games through WoW and Civilization Revolution. Eventually we were playing WAR together and stuff. Good times.
I've found that beat em ups never work. So even though I love Guardian Heroes and Gunstar Heroes, the ladies don't. But it really just depends on the girl in question's taste; the first woman I mentioned got into fighters, WRPGs, and tower defense because she's enjoys the cerebral. She likes thinking about how to counter her opponent's actions, how to force them into the corner, and now that she's into 3S, how to bait parries. She likes roaming a world, talking to the other characters, building the relationships, and killing baddies with the character she built. Tower defense is the same; it's fun for her to build defenses and watch them work. The second that I mentioned loved board games and the sense of being in a different world with other people.
Don't give up! Just observe her personality, pick out games you think might appeal to her, and play them or at least try the demo. If she likes clothing, let her check out something like Resonance of Fate (which I frankly love for it's barbie aspects), if she likes music, maybe Rock Band. If all else fails, most folks will play a few rounds of Mario.