Do girls actually play these games? The only people we know with them are pre-teen to teenage boys.
You can tell from my DISboards that I really like video games! (no, not head games...can't stand those!) I play those instead of watching cable...yuck.
Yes, girls and women play video games. I'm 40 and own a PS2 and PS3 and play them daily. Not to excess, just that I do that instead of watching television.
I like what a couple of PP's wrote about sitting down with your dd and asking her about this purchase. You said "She has never played one before and now it's all she wants." Why the sudden interest? Is it a "keeping up with the Joneses-type thing? Or a teen-status thing? Phooey!
You also said "She spends so much time on the computer and iphone, one more distraction is not what she needs." I'd be asking what kind of grades she gets, does she have a job, do sports, other commitments, and if she gets a game console is it going to either gather dust or suck up too much time that should be spent doing something more productive? If she's a kid who keeps the grades up, honors other commitments with a job, helps around the house, etc. I'd sure consider it.
Gaming is a lot of fun. I play alone, no online and all I bought was the console and a couple controllers + some games. Note that games can be purchased used for sometimes much cheaper and the quality is just fine (and if it's not, the store should take them back - ie. GameStop, I believe, has that seven-day "try it out" return policy which I've only had to use once in all these years).
I've never played an XBox so I'm no help there but other posters have addressed that. Also (speaking for myself), if I were a parent I'd monitor what my kid played...yes, even at 18, although by that age I'd hope to have trained them right. There are a number of "mature" rated games out there that no kid of mine would need to be playing.
If you want to get one, maybe have her look at what games are available on each system and then purchase a system accordingly. Personally I enjoy the "Uncharted" series, "Prince of Persia", adventure stuff like that. And yes, the Blu-Ray feature of the PS3 is a nice addition.