...and no, I didn't muscle it - apparently the seal failed on that side, water and sand got inside, the bearing cage disintegrated, and it cut through the cup! Good news, it cut through so that the threads for the most part were protected by the remaining piece. Bad news, the remnant is still in there and doesn't seem to want to budge easily.
Any good ideas how to extract this without destroying the threads in the bracket? This is an old but very serviceable bike (1992 Trek Antelope) that my son uses, so I would certainly invest in a cartridge for the thing if I can get the bracket cleared out. I'm sure it gave all sorts of warning before it got this bad, but he's not really tuned into machines.
Help?
Feed a hack saw blade through the BB shell and CAREFULLY cut the remnant of the cup in two places. If you can, try to cut through the cup about 95% of the way to avoid hitting the threads on the shell. Then give the pieces a couple taps with a hammer and they should break out. Worst case, if you do cut a little into the threads on the shell, it shouldn't really hurt anything. Might be good to have a shop chase the threads before you install a new BB.
don't think you will cut through a cup with a hacksaw, they are hardened steel.
Use a small punch with the end ground to a chisel point, find a place on the edge of the cup where you can seat the point and tap it around with a hammer, lots of patience needed but usually works.
Note that if you have to resort to more extreme measures and damage the thread, you can buy threadless sealed units.
I don't quite understand what your left with.
Does this idea help?
http://www.sheldonbrown.com/tooltips/bbcups.html
Scroll down to the "home made tool" section.
If your are just left with the threaded part then you could try a rough file inserted into the remaining part.
Push the file into the hole and the teeth on the edge of the file grips the inside of the remaining part. Then turn the file in the correct direction to undo.
(use an adjustable spanner or molegrips on the file to give leverage.)
Ride hard or ride home alone!
OK, this turned out to be easier than imagined. After trying to turn the part with the tip of a screwdriver, it started to move! A little tap with a mallet every so often and the part turned out. Threads were mainly fine, and I was able to thread in the new cartridge with little difficulty.
Thanks all for the advice. It gave me the courage to go after it.