Personally I prefer fleshed out characters, the story line is a big deal for me. I want to relate to the character, also. But I think that is true of most people. Tabula rasa characters bore me to no end, but that isn't gender specific.
What troubles me is the backwards trend with female characters. The new Metroid and Tomb Raider, (though the creator for Metroid is the same dude.) Metroid Prime was much better, Samus was really badass in that I think. And then you have this new one where she's all "omgz babies and how can I faithfully serve my male superior even if it means my death!?" Come on!
Sometimes I feel like developers are actively attempting to drive women away. I wonder if some of them are the same angry gamer dude types that caused all this trouble with Gamergate. I suppose some of them would HAVE to be.
Also a big plus, not "developing" a female character by victimizing her. That seems to be a quick fix for developed female characters that I just don't like. I want powerful female characters. I want the same tough, unbreakable heroines that we see with heroes. I don't think it's too much to ask that this happens a decent amount of times.
BTW to combat misinformation posted previously, Here is a link to the demographics for candy crush saga. While Bejeweled is actually played much more by women, candy crush is somewhat evenly spread. I'm just going to start asking for sources on some of these claims as to what women gamers play.
http://developers.magmic.com/demographic-breakdown-casual-mid-core-hard-core-mobile-gamers/Looking at CoD, there's almost no women playing. But there are definitely plenty of hardcore female gamers, and I think they'd jump on board if the marketing teams weren't busy pretending they were in Madmen and actually tried to appeal to that base.
Social gamers aren't really the same audience as hard and mid core gamers, anyway, so if you say "oh well it's mostly women that play bejeweled" you would be right, but do most women gamers play bejeweled? There's a difference there. (I don't have numbers on that, post them if you do.)
That said, once I'm interested in a game, personally, I don't always play the female character. I haven't liked the skills for any of the female borderlands characters, and have played Krieg, Salvador, Wilhelm, and will probably play Brick when I go back and play the first one. Krieg turned me on to Borderlands 2, so I guess more games would appeal to me if they added more mentally ill characters in them, if they, for some reason, don't want to include women. But that's just me.
FPSs are also kind of a difficult thing to get into. A lot of the men I know that play have been playing them since their teens. I briefly played Borderlands 1 as my first, then delved into 2 as my first real experience. While that is a seriously awesome starter FPS for someone unfamiliar with them, trying to dive into CoD with no FPS experience is just not fun. Team Fortress 2 is another good starter FPS if you don't mind dying a lot. (I play a spy anyway so it is expected.)
I realize a lot of this relates to me personally, and not all females, but I am also curious what other women think about this stuff. What appeals to you? Do you play or want to play FPSs? What kind of games and characters do you prefer?
Developers tell us we should want to play games that involve fashion and ponies, and that expectation doesn't seem to change when we grow up. I'm an adult, I play games to murder stuff or get engrossed in a story. Not brush and preen a horse for 30 minutes.