I talked to the previous owner, who apparently opened up the sub to take pictures of the electronics inside. They suggested that I open it to see if any wires or insulation might be coming in contact with the speaker. Sounds reasonable. While I couldn't recreate it last night, it has happened on more than one occasion, so I think I'll open it up and do a little exploratory surgery.
I had an SVS PB-1000 down there before the FV12, and I was surprised at how much better the old FV12 was than the SVS. I'd hate to have to go back to that. So it's either fix this one up or buy a newer one.
Thanks for all the help and suggestions!