Hi L.,
I recently got back into facebook and set up a page for my 10 yr old daughter. She LOVES it and the biggest thing she'll do on there is Farmville, even my 30 yr old sister is hooked on that game. I have a lot of neices and nephews close to her age who also have a page and she communicates pretty much with them only.
She only has about 20 friends on her list, some from school but mostly family members. I see nothing wrong with it, mostlly because there's so many security settings you can control and you can make her page completely private from anyone but her friends. Even her friends friends can't see her page w/out accepting a new friend request.
Now my daughter did want me to set her up a myspace page back in the day but I didn't like that as much as facebook. Facebook just seems more innocent IMO and has better privacy settings.
I think 13 is a fine age to let her join and just set everything on her page for private for non friends, check up on her page every day (there's NOTHING wrong with doing that) if she wants a facebook, then she should know you're going to check up on what she's doing and monitor her friends. Let her know you want to be aware of any new friend requests and approve before letting them be on her page. And you can always go in there and DELETE any friends you don't approve of at any time if you don't feel their being appropriate. It's very easy to control and my daughter knows that I know her password and will at all times and she's fine with that. As long as she's just communicating with some friends and family, it will be fine. If at any time you feel it gets out of control, just delete the entire page and tell her she has to wait a few more years.
I think she'd have a lot of fun with it. My daughter mostly does the games and makes silly faces out of her family members pictures that always crack me up. It's reallly cute. Take care!