The button in question is the Track Down/Tuning Down button, and now it works just fine, but the corresponding Track Up/Tuning Up button also performs the "down" function (rather than the "up" function).
This has me puzzled as to what could cause such a thing. If I remember correctly, it did this before I did the button fix, so I didn't cause it.
Apart from bad programming in the MCU chip, does any body have any idea what might be causing this, and whether it's fixable.
This is what the unit looks like...
 
I found a schematic here.
I found another schematic here. (The file is a DJVU file, so you'll need this reader to view it).
If you have a paid membership at nodevice.com (which I don't), there is a service manual here.
You can download the regular user manual here.
I notice in the manual that the higher end versions of this boombox come with a remote control, so I wonder if it's possible to hook up an IR receiver to make my unit remote control-able.