Little Claypan Oodnadatta Town Common
The camping options in Oodnadatta are a bit grim. There's the "Tourist Park" behind the Pink Roadhouse, but when we were there it was hard baked earth and pretty unappealing.
Adam Plate at the Pink Roadhouse pointed us in the direction of the Little Claypan. It's in the south east corner of the Oodnadatta Town Common about 5 minute's drive from town (if that). It's a miniature claypan with little dunes surrounding it. A perfect camping spot. We spent a night there, great campfire and a really quiet camp. the kids loved it and all the local wildlife.
You wouldn't want to be caught here when it rains. We had a short sleepless period listening to distant thunder and wondering whether we should get up and move on. It didn't rain and we'd go back to camp here again if ever we were in Oodna.
The claypan you want is in the bottom right hand corner of this map: http://www.pinkroadhouse.com.au/Pink/graphics/docimages/oodnatown.gif
The position of the pin on the forum map is approximate.