OK, so I opened her up and did a little surgery. I checked the ends of any disused PCI-E or hard drive power cables to make sure they weren't touching anything or coming too close to the case walls. Used cable ties to bind them into places where they weren't likely to cause any trouble. Checked for cracks in the cables but didn't see anything. The new case has fantastic cables and cable management options - it's
this Silverstone case, which I pretty much love apart from the drive cage, the placement of which makes it impossible to mount large CPU coolers - even towers - with the the fan blowing on cooler. I have mine pulling air thru the cooler, which works surprisingly well, maybe because it blows straight out of the case.
So I also checked the fan cables and got those up and off and away from the case bottom and walls. I checked all the drive cables. Didn't see anything odd.
Then I remembered that mPCIe card at the back of the board. I recalled thinking when I installed the thing that it has lots of little metal solder points that press right up against the backplate. So, I stuck a piece of electrical tape on a zip tie and shoved it down between the backplate and the mPCIe card. Don't know if it'll make any difference, but if the problem does go away that's the likely culprit.
We shall see...