We have a Sony Cybershot and we love it. We use it as a family. My daughter is almost 14 and she has asked about her own camera.
She has been using our older model Fugi which is a great digi cam as well. The downside to the Fugi is that we keep putting batteries in it and if you travel, you have to make sure you take plenty of batteries. Other than that, it is a great camera.
We did spend more than you would probably want to for a 6 yr old on the Sony. I do like the recharable battery that comes with it. We bought an extra battery so that we can always have a backup. We also got a larger memory card. I love teh Sony.
My suggestion is to go to Best Buy. The staff is knowledgable and can guide you to the right camera. There are many cameras out there in different colors, etc.. I am sure there is a reasonably priced one for you.
The main thing to NOT do is to go cheap. Buy the best product you can afford that will last her a while. We are big Best Buy fans because we have been able to get good quality advice and products (dishwasher, fridge, computers, cameras, wii, gaming products, etc) from them and they stand by their products.
Enjoy!