Depends on what type of girl she is.
I would say that perhaps the most effective choice would be an MMO, simply because they offer so many different play styles, such as pve, pvp, tradeskilling, economics, social, ect. as a catch all.
If shes into narrative, go the more narrative route with an RPG
If shes more action oriented, try the shooter route.
If shes into crosswords or sudoku, try something puzzle oriented.
Its all about matching a game style with her other normal interests.
But I honestly suggest, that if you dont know your girls personality well enough to know what types of things shes into that your turning to a forum for advice, perhaps you should ditch the games all together and focus on getting to know her first, once you nail that down, figuring out what type of game style matches her interest shouldnt be all that hard.