How can I change the desktop wallpaper for a restricted, parental controlled account?
You can set the wallpaper to automatically show images from a certain folder, switching images every X amount of time. This way, your son can put the image he wants in the proper folder and remove the old picture, and the next time it is changed it will choose the only picture there.

Use the + button to select the proper folder, change the stretching rule to fit your needs, and check the box for automatically changing the picture. If you don't set it to automatically change pictures, it will look for the last picture used and show a blank screen when it doesn't find it. You can force the image to be reloaded by logging out and back in, or by executing killall Dock from the Terminal.