Ok, found the problem and eliminated it. vpamicue was right - i went to PopupBlocker and found the Mozilla window listed in its block list. How the hell did it get there? Why didn't I think of that myself? Arrrghh!
Still, it was hard to find because of all the secret data files Windows keeps. Registry, javascript, java, javavm, ActiveX controls... any of those could have been to blame.
This family PC uses XP. I "repair" the XP installation (which is actually reinstalling, lets get this straight) only as a totally last resort. Why? Because it (a) almost never fixes problems like these and (b) always breaks lots of stuff, including the VIA drivers its taken me ages to get stable.
Well anyway, thanks to everyone for their suggestions. I can genuinely say that if it wasn't for AO coming to the rescue again I'd still be looking at this computer and scratching my head.