I hate being asked to recommend something because a guy is 'trying to get his girlfriend into games'. This, of course, mainly happens when I don't know and have never met said girlfriend.
It may come as a surprise but a person's gender doesn't dictate their interests or tastes. I should hope that you know your girlfriend more in-depth than "she's a girl" which is currently all I know about her. Somehow you think that's enough information for me to know her likes and dislikes. Somehow you think my magical girly perspective will let me know what a complete stranger will find entertaining. *Spoiler* it won't.
Why is there a games based blind spot with this? If you're asking for a film or tv recommendation you add details of past interests. You let the person know if your girlfriend likes Game of Thrones, Marvel movies, Buffy or Sex and the City.
"Oh!" But you cry, "my girlfriend hasn't played any games!" A person's interest in games isn't completely separate from their other interests. Did they love Stranger Than Fiction? Think the narrator in George of the Jungle was great? They'll probably like The Stanley Parable. Do they think Patricia Cornwall is amazing and are obsessed with The Wire? Try Her Story. See where I'm going here?
This isn't to say I won't recommend games at all. But I won't recommend with the sole knowledge that she's a she.
If you can come to me saying your girlfriend is a nurse. She hates superhero movies and gratuitous violence and loves Stephen King books and Friends. And, mostly importantly, you've suggested games you like but she isn't into them. That I can work with. That's a person.
If you come over saying "I have a girlfriend, what games would she like?" Please take a moment to think. You actually know this person. I don't. You should know what she likes. I've never met her.
Stop acting like wearing skirts and lacking a Y-chromosome somehow means we've a telepathic link.