It depends on the child and where you live ... in TX, the "prestigious" overnight camps have waiting lists years long. For some camps, they need to get on the list before they are born. Others are not so bad. In GENERAL, the private overnight camps run about $700-1K per week -- some more, some less. Girl Scout and church camps are MUCH less.
I went to campfire girls camp (like girl scouts) and I can honestly say that it was the very best experience of my childhood. I learned so much and made great friends. I wish I could have gone for weeks at a time. otoh, my sister went to the same camp and was miserable!
I was a counselor at a camp outside of Branson, Mo (Kanakuk), which was an awesome place. At the time, their sessions were 2 or 4 wks. I had cabins of 7 yo girls and they ALL did fine after the first couple of days.
My dd (now 9, in 3rd grade) went to a private camp in the Texas Hill Country (most of the private camps in TX are located in the hill country) last summer for 2 weeks and LOVED it. Her camp's longest session is 3 weeks -- most range from 1-6 weeks -- and she begged to go for 3 weeks this year. We are taking an extended vacation this summer and can only schedule 2 wks and she is bummed. My younger dd7 is joining her for 2 wks.
In my personal opinion, to get the most out of camp, a 1 week session just isn't long enough. The kids can hang on to the homesickness if they know it is 5-7 days. But, if it is a 2 week camp, a child who is ready for camp will get over the homesick feeling at day 3 or 4 and join in the fun. They won't be ready to leave at the end. (I'm not saying to send a child who isn't ready or doesn't want to go, though).
At most camps, parents cannot "chaperone" at all. It seriously increases homesickness. But, most camps now have daily updates and post pictures (via a secure website) of the events of the day. It is reassuring as a parent to see a pic of your child having fun and smiling w/o you!!
As far as particular camps, do some online research for your area. I could rattle off 10 Texas camps, but there aren't tons of out of state kids there.
Check Kanakuk if you are a (relatively) conservative Christian. It's a great place. Find out what your dtr wants out of camp. My dd would only consider a place that had riding every day, so that narrowed our choices considerably. We visited camps the summer before and during the year. My dd picked her camp herself and, luckily, it was the one I wanted, too!!! I think that helped tremendously in her attitude. If there was something bad, she couldn't blame it on me and get cranky and homesick!!!
Good luck in your hunt!!! Camp can be a fantastic experience for your child if you find a place that suits you and your child!!