Amazon.com is selling Uranium Ore (amazon.com)

Wouldn't use uranium, because as another poster mentions it's more a novelty and there are better materials you could use, but you could make your own cloud chamber to observe radioactive decay with the naked eye. Makes for a really cool science demonstration :)

http://www.youtube.com/watch?v=pewTySxfTQk

http://en.wikipedia.org/wiki/Cloud_chamber

Test your geiger counter, it sounds like. I think that's about it.

Make a hardware RNG that's truly random by detecting the interval between decay events :)

There isn't a whole lot you can do with it, you are basically getting a rock. You can show it is radioactive by running a geiger counter across it. Interestingly enough the old rare earth mine in Nevada, back when it was closed and essentially "abandoned" had a variety of radioactive materials in its tailings (the most common being Thorium) we did a science trip once to grab some and figure out the various isotopes in it.

If you put it in an air tight container you can collect Radon.

That is great stuff! :-)