@Paul_Haskins @JDRoberts @ritchierich. Thanks for the input and ideas. After I posted this, I did a lot of Google searching and indeed found this is a common problem.
As a first shot at solving it, yesterday I changed the Z-Wave switches out for some simple timers that have photo cells and an on time delay of 2, 4, 6 and 8 hours. I selected 6 hours, so the lights will go on at sunset (about 4:30 pm late November) and off 10:30 pm. This will help narrow down whether it’s the Honeywell / Jasco Z-Wave switch or a more general fault of water ingress to the lights or cord connections between light strands. They worked fine yesterday, but it was a dry day. Rain and snow are in the forecast for Friday this week – that will be the big test.
I just talked to an electrician and he says GFCIs have gotten a lot better in the past five years and even improved a lot the past two years. The new ones are much less susceptible to nuisance trips like I’m having. He likes Leviton. Next week, I’ll swap out the old GFCI for a new one to see if that helps.
If I continue to have troubles with the Honeywell Z-Wave switches tripping the new GFCI (and assuming the new timer with solar cell works), I’ll install a power monitor on the GFCI outlet in the garage to send me notifications when the power goes out on that outlet (i.e., when the GFCI trips). Unfortunately, the Z-Wave Power Outage Sensor by seven7express is no longer sold. But its simple to use a 5V solid state relay hooked to the dry contact of a Fibaro window/door sensor. The 5V will hold the NC connection open when power is present. When power fails, the NC will go closed and I’ll get a push or SMS message that power has failed. See “Power Loss Alerts.”