She's at an age where she's very socially-minded and very impressionable. Martial arts aren't really the "cool thing to be doing" these days...not even in Japan. Best bet is to find a dojo with girls her own age, or just a few years older, that seem cool, popular, attractive...someone your daughter will respect, maybe even admire, in regard to the value set of a 13 year old girl. In my dojo, when someone wants their daughter to train, I arrange for them to meet a girl at our dojo, Kathy. She's maybe 15, a black belt, very mature for her age, and has been in local and inter/national newspapers because she won a gold medal her first Pan-American Games tournament this year (sparring, karate). She can be really intense for a 14-15 year old girl, but she's also very girly once you get her off the mat, and very sociable at her school. She can serve as a defacto role model for young girls considering training. Doesn't always work that way (girls often seem to find reasons to hate each other), but its a solid foundation.
Biggest thing is to check her "street smarts." No amount of training is going to make any young girl invulnerable. Probably the best attribute of training in a traditional martial art would be to give her confidence and independence. Ensuring she knows her limitations, what's out there, that help positively effect her decision making will be more beneficial than learning a wrist lock or punch. If she's out drinking at a party (underage), this is already a bad situation...but hopefully when one of her schoolmates meets an older college guy and wants to go off somewhere with him in his car, something will click in her head that will say "this is really f-ing stupid, I'm not going."