Thx Vincent. Actually when I wrote it, I did mean <Alt>Space and that's what I tried unsuccessfully. On previous occasions I've had two issues with setting shortcut keys in Gnome. The first was when I had Compiz running and it superseded whatever I had set in Metacity. This was easy, just set the keybinding in Compiz. The second problem was that some shortcut keys change immediately, but a few are resistant and require a log-out and log back in again. I knew about this, and tried it with MintMenu, but it didn't work. At that point, I posted. I had NOT tried rebooting (as that's rarely needed in linux). Finally I tried that and it worked. I actually used <Alt>space (lowercase), but I don't think that makes a difference. I'm a bit embarrassed.
Maybe MintMenu needs the reboot because something it needs is loading very early in the boot process. I don't know, but reboot did it. Marking the first post Solved.