Using your mouse to click the menus works OK, and you can highlight the text using the keyboard.
This symptom was unique to certain people's profiles on this machine. Others logged into this machine did not have the problem.
I'm not totally sure what fixed it. I was trying the options listed in this Microsoft Support article (link below) including the Fix-It button and after a couple of steps would try it again.
I think the cure was deleting all template files (*.dot) from this directory:
C:\Documents and Settings\username\Application Data\Microsoft\Templates\
I was about to do the step regarding COM Add-Ins (have to open Word for that) and viola it was working.
I had already tried deleting the normal.dot file from this and other folders, but that didn't help.