Yes, I took DD for 2 nights alone last year, aged 10 (DH swears that he has passed the age for sleeping on the ground). I hadn't camped for 17 years, until I went away with the Cubs the previous year. DD had done 3 camps with Cubs over 13 months before we went "solo", and we went to a site which we knew was quiet, had decent wash facilities and allowed us to light fires (Scout site in a quieter time!).
She loved it! Our cubs do almost no pitching of tents, but DD had a reasonable idea of what to do so held things, and knocked in pegs with the mallet etc.
She went off to get firewood more than me! Although I tended to carry the water barrel when full.
She really adored the freedom to wander around the site, climb trees, gather sticks to start the fire, paddle in the lake etc. We did a long hike with a swim on the middle day, played cards in the evenings, and 1 night she was "adopted" by a Venture Scout unit and taught how to make dreamcatchers (while I sat and drank tea with the leaders, as per usual Scout Camp tradition!).
The hard part is pitching and striking camp, so a 3 night trip would be better than 2, and I definitely wouldn't just go for 1 night with a tent (campervan would be different though!). Make sure your sleeping area is warm enough (insulate under the sleeping bags as well as having blankets for over the top in a tent), and enjoy the adventure! I agree about keeping food simple, and always bring a pack of cards, and maybe a favourite board game/book etc.
I am currently in the process of planning another outing, either 2 or 3 nights, for another week towards the end of this summer, with DD. DH works very hard and travels away a lot, and we will have a family holiday earlier in the summer, but this is a nice extra "girls adventure" for DD and I. (And maybe her Godmother).
(For ideas of how to camp, set up a campsite, what gear to bring, easy food ideas, outdoor skills etc, there are loads of great resources on the web - adventure stores, Scouts and Guides, and loads of YouTube videos as well!)