Well, I'd like to... What would it take to space harden an AVR? Or do space hardened AVRs exist already?
You are starting in the wrong end. First build prototypes and explore ideas, get stuff working, then figure out how to get it to work in space.
The cooling would be simple enough, just attach every single thing to a heatsink (probably including passive devices like resistors and capacitors).
Space is cold and any heat produced will radiate away quickly. I don't think cooling will be an issue.
What I want is to be able to build a cheap robot with a camera that could go to the Moon or Mars and wander around taking pictures, the Moon would be preferable.
Despite that I agree with most people here and find it highly unlikely that you will be able to actually send stuff to the moon, this sounds like a really cool project. There are lots of interesting problems to be solved, but radiation and heat are not the biggest ones. You should focus on building prototypes that works on Earth first. That will keep you occupied for a long time. When you have a robot capable of wandering around the desert without any maintenance, doing all the stuff you want it to do on the Moon, then you can start working on making it "space-proof".
My conclusion so far has been that the reason space-based robots tend to be extremely expensive (in the range of billions of dollars) has less to do with what they make them out of than how much it costs to get there. In other words if it costs on the order of trillions to get there then it doesn't really matter if you're spending $1000 on your robot or $10 billion, it's such a tiny part of the overall cost anyway, might as well make it worth it.
If it costs trillions to send your robot to the Moon, you'd better make sure that everything works really, really well. That's why it's expensive. Fixing a robot on the Moon is not easy ;-)