Time for a follow up...
While I wasn't looking they got an air conditioning man in to fix the air conditioner and now it is working, turns out that the wire that burned was not the wire going to the compressor, but a wire going to their old wood burning heater. Apparently if they didn't cut every wire in their closet when the fire happened the air conditioner wouldn't have stopped working.
The wire that caught on fire was a modest gauge romex that IIRC was wired directly into the electric heater/fan in the closet, not to the breaker panel. The breaker for the central electric heat unit may be more than 20A (guess, probably 240V). Apparently this wire going to the blower for the outdoor wood heater was not properly tied off, when the heater was removed (by a non-professional DIY) sometime last year.
I do not completely understand the cause and effect time-line that said the fire occurred coincident with the air conditioner being powered on. It appears there was a quasi-short across the improperly terminated romex that was enough current to char it, but not enough current to trip the probably pretty robust breaker on the heater. I did get three different stories from 3 different people about the event so the fire may have been random waiting to happen, when a butterfly flapped it's wings nearby.
The burnt wire is still unfinished business that they plan to have a real electrician replace (air conditioner guy didn't do electrical work). They have unused breakers in the sub panel, so can wire some proper gauge wire into an actual breaker out to their shed. I strongly advised using a more robust wire than was in there before, but hopefully a real electrician will not repeat the previous bad behavior.
I feel slightly better about the situation as they dodged an expensive repair to the air conditioner, and appear on a path toward a professional electrical repair.
JR