Well, I've now logged how many clicks my Geiger counter saw each minute since some time on May 20. It's more than clear that the click rate is falling gradually. When I first became suspicious a while back, the rate was about 50 clicks per second and now it's about 45. The radioactive source I'm using is the little dot of ²²⁶Ra stuck right up next to the GM tube that I've used for some time now. Here's the data:

As with earlier charts, the blue line is the average of the whole run up to that point and the red line is an exponentially decaying moving average with a decay period of an hour -- a measure of the behavior during the previous hour. For my purpose -- building a timebase -- this is not good at all.

I'm relatively certain the problem is with the long-term stability of my Geiger counter, not the physical setup or the radioactive source. It's still possible, I suppose, that there's an environmental problem (e.g. variability due to Rn), though there's no obvious diurnal pattern to the data or correlation that I can see with the weather.

Time, I think, to look into a different (type of) detector to see how it behaves over time. I need to do some research on the alternatives.