Facebook is a social networking site. Let's think about what that means for a minute. It's a hangout, kinda like a bar, and there are all kinds of people, and the mechanism is already set up for any of the people using facebook to contact you (or your daughter) - and that includes all the unsavory people that you most definitely don't want anything to do with, as well as people who are very sweet and nice and tempting in all the wrong ways.
So it's not just a matter of trusting yourself, or your children to not do something wrong. What will they do if someone from the opposite gender wants to be their "friend?" Totally innocent, just wants to have a lot of friends so they'll look popular. but once you're friends, maybe they'll send a message saying thanks...and it can all go downhill from there.
Reb Ano - I agree that clearly, the girl has issues in the relationship with her father. but here's the thing - she has a whole community of people with whom she can comfortably share the most intimate aspects of her life, which she ordinarily would have trouble telling her closest friend. And depending on the situation, you have a whole community of angry teens encouraging her, and tellign her that it's GOOD for her to be angry at her father.
Sometimes, a person needs to be able to talk about things, and a blog can offer a great outlet. But all too often, it serves as an easy platform for people to air their anti-torah views (though of course they claim that they're supporting the way that judaism is "supposed" to be, unlike the way all those old rabbis think), and for others to support them. Kinda like porn online - it was always around, but now, with the internet, it's easier to access, so more people get involved.