My fiance is very, VERY picky about what kinds of games she'll play. I asked her what she likes in video games and this is the literal criteria she gave me:
-Must be able to make a female character, and customize it
-Must have good, non-cartoony graphics
-Must have fun, likable companions
-***MUST BE AVAILABLE FOR XBOX 360***
-Medieval or Fantasy setting is preferred, Sci-Fi is generally frowned upon (Mass Effect is the only exception she's ever made, and it lost points for it)
-Romance arcs are a plus
Some of her favorite games to date are Skyrim, Dragon Age (Origins, Awakening, and 2), Mass Effect (1, 2, and 3). She tried Fable and disliked it pretty intensely for the graphics, saying they made the game unenjoyable for her.
Low-cost games are ideal, but newer ones are welcome as well. If they have available demos that's even better!