I was thinking I might have to do an archive-and-install, which would have been a pain, but then I remembered seeing Apple's article on how to change a user's short name, which involves copying the files from one home folder to another. So I created a new user 'test' and got to work.
Here's how I solved it, while logged in to the troublesome account. In Terminal, I ran these two commands (where myuser is the short username for the troublesome account):
$ sudo cp -R /Users/test/Desktop /Users/myuser/Desktop
$ sudo chown -R myuser /Users/myuser
This would, of course, delete anything left on the desktop if it wasn't empty, so use with care (have you backed up lately?). I could probably have used cp in Terminal to copy things out of the desktop first, if it wasn't empty already. I'm not sure what caused this problem, but this has solved it. Possibly it was some sort of permissions problem, or the Finder was seeing the desktop as a package from the failed copy command.

