I bet anyone that says that that OP is lucky that they have a gamer girlfriend has never asked their girlfriend to play with them.
My girlfriend is techoretarded and I asked her if she would like to play World of Warcraft with me for about a week. She picked up the game easily and enjoyed it for the trial then abandoned it because she didn't want to get addicted to it...like me. Even crazier, I thought she would want to play Alliance, but she chose horde and was a troll.
TL;DR, if you ask your girlfriend to play a game with you, she will. You just have to find the right game.