The HV transformer. Power it in a dark room, you'll probably see it spark.

So I just kept cycling power and... it has been powered on for over 2 hours
now with no problems.

I'm beginning to suspect the electrolytic caps. Being el-cheepo (no-name)
brand, they probably went intermittent short or low ESR and blew the fuse.
After replacing the fuse (remember it's been over a year since this thing has
been powered on for more than a few seconds at a time) the caps needed a
little time to reform.

Or maybe a power surge popped the fuse (and the cap re-form thing still
applies).

I think I'll replace the caps -- for peace of mind -- and see if it continues
to work.
--
John English