not looking good
After opening up the relay it didn't take long to see what the problem was. The coil was in great shape which is why it was clicking over, but the contact was well burnt up.
I found out that Ducati isn't like BMW at all when it comes to old parts, they're more like H-D. Only worry about the new stuff, and anything older than 5 years or so will be relegated to the salvage market. I wound up finding a used relay for $11.95 on eBay (with a video showing that the bike was working before disassembly), and then an automotive generic relay on Amazon for $4.20. Since the Amazon relay is lost in the mail currently, and the old Ducati one happened to arrive today, I decided to try that option.
After a brief recharging of the battery, this 94 is back to life. Now to tackle the front suspension, rear brake pad change, and oil change, and this bike will be back on the road again. Just listen to that Ducati clutch clang around...