Interestingly, I tried again today and it worked. Very odd. I had tried at least 10 times previously (different times/days, different browsers) to no avail.
I don't know how much more detail I can give. Submitting a screen shot would have been helpful, but I don't see a way to do that.
I was logged in as a SysAdmin, went to the System Administrators page, checked a box next to one of the Usernames, then clicked Edit. Every time, regardless of which user was chosen, a message would pop up saying "No item has been selected."
As I said earlier, the problem is not occuring this morning, so I guess all is okay.