I went to the doctor with this problem. He said it is osteoarthritis of the thumb joint, and very common. If you Google, you will find more information about it.
What I finally figured out is that repetitive motion was causing the immediate problem. To make a long story short, I stopped playing Wii games with the remote controller. It probably wouldn't cause a problem for most people, but my thumb and little finger bones happen to be half of normal size, so operating the button on the remote with my thumb was putting stress on the thumb joint. I purchased a thumb joint splint from Amazon and wore that every night, in addition to giving up the remote controller, and my joint has definitely gotten better (although not perfect). Here is a link to the splint I purchased:
http://www.amazon.com/gp/product/B00...00_i01_details
(I leave the velcro in place and just slip it on and off my hand without readjusting the wrist velcro.)
For osteoarthritis in general (I have it in my knees and spine also), I take calcium citrate, glucosamine chondroitin, and Vitamin D every day. I started this after my knees got worse so rapidly. It seems to have halted at least obvious progression, as my knees have been stable for the last 8 years or so. I'm assuming this might help my thumb joints too.
One other thing I take every day is citrus bioflavanoids. This seems to help with my carpal tunnel syndrome (diagnosed as moderate to severe, now in remission). May not be relevant to thumb joint problems, but I figure it can't hurt.
Edit: I want to stress that it's important to have a doctor diagnose this condition. After describing the problem, my doctor was able to place pressure on a specific spot to determine whether or not it was osteoarthritis. There are certainly other medical issues that can create pain or discomfort in the thumb joint.