Okay I was premature in claiming victory over the clicking. The constant clicking between day and night modes reappeared.
The good news is that after endless resets and different firmware versions I believe I have found the cause, which is not related to firmware versions. I can start and stop the clicking by changing the image contrast setting.
Details: The camera is pointing out a window at a fairly high-contrast natural setting. With the image contrast set at 0, the camera starts clicking and cycling between day and night modes. When I lower the contrast to -5 it stops. Once the clicking stops, I can raise the contrast back up as high as -2 without it starting again.
The brightness setting doesn't seem to matter.
Hope this helps!