Beware of little girls and a focus on vanity and appearance. Both my daughters are pretty, like to shop and care about their appearance. That being said I am dismayed at what seems to happen to girls around 10-11 years old, at least where we live.
With middle school comes an extreme emphasis on looks for many girls. They start wearing skimpy clothes, fake nails, extensions and yes, dying their hair. They spend a lot of time on maintaining their look and revel in the attention they get. At the same time it seems boys are getting "noticed" for their good grades or athletic achievements. It isn't all girls but it's a fairly large percentage of them and it makes me sad.
At nine your daughter isn't quite there yet, I'm sure starting to dye her hair doesn't seem like that big a deal and it really isn't. Just be careful of the message you're sending about what's really important.
At nine Is SHE really concerned about her hair being flat and stringy? If so do you want that to be something she focuses on? The message is your hair isn't good enough the way it is, you need to spend time, money and energy on making it look better. Between hair, nails and clothes it seems we are expecting our little girls to spend a lot of time on appearance and that has to come at the expense of other pursuits.
I have a ten year old who does like to paint her nails and curl her hair on occasion, it is fun to be a girl. Honestly, when she's up on the pitcher's mound in a dirty softball uniform, hair in a ponytail and huge smile on her face we find her more beautiful than ever.