The symptoms:
Occasionally, the music will either stop playing or get hung on a single note. When the music changes, it usually starts working again. For example, the music will stop working, then when the ball drains and the music restarts on the next ball, the music will be back. I have seen "Music Error" when entering music test, but not every time; it's actually very rare that the issue shows up at all. I haven't noticed any problems with sounds or voice.

What I've done so far:
I've replaced the ribbon cables and cleaned the mating connectors with 99% alcohol. I have inspected the headers and nearby traces under my microscope. I have not replaced the headers yet since I didn't want to do board work without identifying the problem.

Any suggestions on where I should look next? I have an oscilloscope, but I'm not sure what to look at. It seems like it's probably a problem with a command that selects what music to play, but I don't know enough about how the music works to know what to look for.

Ok, that was interesting. I started by reseating only the Music/Speech ROMs on the audio board. I played a test game, and a bunch of sounds were missing. I hadn't touched the sound ROMs yet. The sounds were all silent in sound test. They never have been prior to this.

I then reseated the sound ROMs, and they came back, but the spoken ones were garbled. "Uh oh. Looks leiwia reeigh!" I reseated them again, and it was back to normal.

I noticed that the pins on the ROMs looked like they had been sanded, so I'm guessing the previous owner also had issues. It's working now, so I'll leave it alone until it acts up again. I'm guessing new ROMs will be in my future.

Now I'm not sure what I was actually reseating when I did it.

John's site says:
Sound ROM (U4)
Speech ROM Set (U19, U20, U21, U22)

The labels on my chips say:
U4 MUSIC/SPCH
U19 MUSIC/SPCH
U20 MUSIC/SPCH
U21 SOUND ROM 1
U22 SOUND ROM 2

U4, U19, and U20 are the ones I reseated during the first round. U21 and U22 are what I reseated once sounds started acting up.

Yeah, that's what I was just searching, trying to find out how often people have to replace the sockets. I've already installed NVRAM, and it's the oldest board I've done that on. The board is fragile enough that I don't really like working on it much.

I decided to pull all the ROMs, clean the pins, then reinstall one at a time and note which sounds correspond to each chip. Here's my findings, in case it's ever useful to anyone.

U21 - Sound test 00-04
__All of these are beeps and other simple sounds
U22 - Sound test 05-07
__05 - Uh oh!
__06 - Looks like rain
__07 - Thunder sound
U4 - Now, I get 4 beeps on startup. No new sounds work.
U19 - 1 beep on startup. All of the music test tracks now work, but music still locks up in game.
U20 - All is working correctly now, obviously.

If it starts acting up again, I'm convinced that I'm going to have to replace sockets next. I do have a bunch of machine pin sockets, but I'm avoiding doing board work unless it's really necessary.

Thanks for the help, guys.

-12V is on the analog side of things, and this certainly feels like a digital problem. I'll be sure to keep it in mind, though. I have completely replaced the ribbon cables.