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Already done: see peltier device. They are already made to the correct size and probably better efficiency.
http://www.peltier-info.com/
http://en.wikipedia.org/wiki/Thermoelectric_cooling

such a device already exists. It's called a thermoelectric generator. Here are some good links: http://www.peltier-info.com/manufacturers.html They're not very efficient, but are very reliable.

I think you're right. This may be a more efficient Peltier by virtue of the semiconductor material. One of the problems associated with getting a temperature differential across a Peltier thermoelectric generator is that both the hot and cold sides need to be electrically isolated. In other words, the best way to conduct heat to and from these devices is to use metal. Most metals are good electrical conductors and if you used them you would be shorting the leads coming off the device. So the trick is to conduct and dissipate heat efficiently without conducting electricity. Not so easy. There are many materials/techniques used to do this. http://www.peltier-info.com/manufacturers.html

Given the increasing temperatures of CPU operation, it should be feasible to use a Peltier device to generate enough electricity to trickle charge the laptop battery or power an auxiliary cooling fan (the hotter the CPU, the more cooling is needed and the more electricity gtenerated by the junction to power cooling.)

This site http://www.peltier-info.com/ offers a pretty good survey of devices already in process for this.

Couldn't one package a lump of radioactive material in an array of these? If I recall correctly, applying heat to these creates a current. How much? Hell if I know, I'm not an electrical engineer :P

This could be addressed with a Peltier cooler. Of course the Peltier has a hot side as well. But we wouldn't be using the Peltier for a cooler for the system -- just for the Stirling engine.

If you hook a Peltier cooler up to a power supply and turn it on without attaching it to anything, within 15 seconds you'll notice condensation on the ceramic part. Within another 15 seconds that condensation will start to freeze. I had a friend who used a Peltier on his AMD K6 CPU years ago. He wasn't much into over-clocking his machine, but just over-clocked it 33MHz or so. After about a month his machine started performing very unstable. We opened it up and, I shit you not, there was mold or mildew or something growing on the CPU! It was great! Except for the, uhh, dead cpu. . . Anyway. . . bye.

I wonder why it is called PeltierBeer btw.

Here is a buttload of information on solid-state cooling and other odd functions of peltier devices.

It seems to me that the neatest, cleanest refrigeration technology would have to be based on the Peltier effect. No gasses, no moving parts, no noise. You can even buy Peltier-based refrigerators already, though of course not at mass-production prices.