I use facebook personally and find it very useful, in the sense that it allows me to contact friends just as easily as email, and share photos, etc. I only ever add people as my friends who I know and speak to regularly.
If your daughter is 13, i believe facebook have set it up so that when you put in your date of birth, under 18s or under 16s have automatic privacy settings to make it more difficult for anyone to see her information. But you can change these anyway. Personally I would let her have the account, on the premise that she tells you the passwords, and that you can check the privacy settings, and also monitor the friends she adds. Set it up so that only friends can see any part of her profile, and perhaps make the profile picture an avatar style picture, as somebody already said.
If you use it right, there really is no problem with facebook, it is very secure. Its just got a slightly worse reputation at the moment because of the idiots who allow the whole world to see their profile details, photos, etc, but this can be easily changed.
I havent used bebo before but believe its more like myspace, which personally I dont think is as secure or useful as facebook.