After sort of not paying attention to how her two-week stint at the avionics shop is going, I got an update from the technician yesterday.
The transponder's receiver was too far off calibration to be retuned, so it had to be replaced with a refurb KT-76A. The encoder turned out to be fine, but the altimeter itself was 40 feet off, too far for IFR certification, and had to be overhauled. That explains the transponder funnies which a lot of us had experienced, but I also wanted them to look at the random avionics power-offs that some pilots had reported, including one report of total avionics failure. I didn't expect them to find anything since they were really transient reports and I hadn't actually experienced anything myself. But I was wrong.
This plane has kind of an odd avionics master switch. Instead of a nice hefty on-off switch which connects the avionics bus to the main power bus, it's a three-position thing which either does the former or, in what's labeled the "Emergency" position, energizes a relay way back in the tail near the battery itself, which connects the battery directly to the avionics bus. This connection, however, as they discovered today, is via a 22-gauge wire running from the battery up to the instrument panel. That's much too thin a wire to power the entire avionics stack, and that wire had been getting really hot, to the point of melting the insulation.
My guess is some pilots had been flipping that switch to the so-called "Emergency" mode for normal flight, causing that wire to heat up and softening the insulation on it. Then one good bump of turbulence, and something would probably short, causing any number of weird things to happen in the stack.
I have to run the maintenance log out there tomorrow afternoon so they can try to sort out how this got installed this way to begin with; I'll get more details from them then. In any event, she's still a few days from being flyable again.