Thanks, everyone, for your responses.
Southpark:
No, the system has never operated for many months without a glitch. Yes, I have made software modifications (enhancements) from time to time. No, the resets did not start at the same time as any of those enhancements. Yes, recently, I have made modifications to the wiring (see the "what I have done" document) and have tried several different power sources. I have also reverted to breadboard wiring in place of all soldered connections (because the TEST system works that way, but the PROD system was having resets). I know breadboards are supposed to be for prototyping, but when the soldered (and re-soldered) wiring didn't solve the reset problem, well - trying something is better than not trying.
Wawa:
In my "what I've done" document, I explained that I replaced the oneWire single bus with three separate buses because the star topography I was using was off-balance. One leg (TANK) was 6 feet, one leg (OUTS) was 15 feet, and one leg (COLL) was 51 feet. Changing the pullup resistance value on the single bus didn't seem to help, so I tried 3 separate buses. OUTS and TANK have 4.7k resistors, and COLL now has 2.2k.
The diode across the relay transistor was an idea I tried after finding it on the web. It didn't make any difference in performance. I'll remove it.
The transistor is a 2N3903 NPN. What value of current limiting resistor should I use on the base?
I'm not sure how I would apply a diode across the contacts of the relay. The motor is AC. How would I do that?
I'm working on the 5v power source suggestion.
Paul__B:
I'm glad the step-down voltage regularor I have is a decent one.
The power requirements for the Uno that I've read on the web have been confusing. Some say one thing, others say something else. You mentioned the Nano in a previous post - is it significantly better for "real-life, needs-to-work-flawlessly" applications? Can the Uno's code be ported directly to a Nano? What differences are there between the Uno and the Nano that might cause me to change wiring, code, anything else? I'm curious.
You said
extra electrolytic bypasses of a few hundred µF across relay modules would be advisable
I'm using a 12vdc coil SPDT relay with 30A contacts - no module. That relay makes a connection between the source side and the motor side of the hot (not common) wire of an AC power source. So how should I connect the electrolytic capacitor across "the module"? Does that mean across the coil? Or does that mean across the contacts (not sure how I'd do that with an AC motor)? Or was that suggestion only for a relay module?
6v6gt:
As mentioned in response to Southpark, currently the system is back on a breadboard. I did that because the soldered version wasn't the answer, and changes are easier on the breadboard. When the thing gets more stable, I'll go back to a soldered board.
See my response (just above) for relay information.
ugfrog