Hi,
Some of the commercial absorber plates use copper backing that is 0.007 inch thick. That might give you an idea.
The purpose of the copper backing is to absorb the solar radiation, and then to transfer it by conduction along the copper to the tube. The thinner copper should absorb the solar radiation just fine, but the thinner the copper backing, the higher its resistance and the more temperature differential is needed to transfer the heat along the fin to the tube. The higher fin temperatures make for more losses out the glazing. If your copper is thinner than the 0.007, you could space the tubes closer together, or just take the (probably small) decrease in efficiency.
Basically, you want the fin to be thick enough to do a good job of transfering the heat along the fin to the tube -- its not a good idea to make it so thick it stores heat -- storing heat IN the collector is almost always a bad idea.
If you can provide the copper thickness, I have a method from the "Active Solar Collectors" book to estimate fin efficiency.
Gary