I did my first Discover Scuba at 13, and got certified the next year - it was one of the best things I've ever done! I have no regrets that my parents let me do that (ok so I might be a little biased ...). Sure, 13 year olds don't think the same way as adults, but most of the time they make amazingly good divers. We frequently get kids in our classes, and I am constantly impressed at how well that they do in our very less than "ideal" diving conditions.
I'm going to agree with the suggestion that you should take the class with her if possible! I think this would really help her stay motivated, alleviate any minor anxiety that she might experience, and most importantly give the two of you something that you can do together! It sounds like this is something that she's really interested in, and I'm sure it would mean a lot to her (and open up sooo many more opportunities for her after certification) if you did it together. I was always slightly disappointed that my parents never really took an interest in diving, and my dedication to it is still a little bit of a disconnect between us.
IMHO doing the classroom portion with an instructor is more engaging and can give a better sense of what to expect (based on stories, answering questions the student might have that aren’t necessarily in the book, etc). Regardless, the bookwork is really not difficult, and most of the 11-13 year olds I’ve seen have had no problem. Definitely talk to the instructor who will be doing your class and get a sense of his experience working with kids. Most reputable shops are pretty good about setting up certain instructors with younger students though!
A lot of people have mentioned having your daughter do a discover SCUBA before getting certified which would be a great motivator, but you might also want to try one and make sure that this is going to be something that you will be comfortable doing!