Can I remove admin rights from an OS X account without using the command line?
Aha — the reason why this option was greyed out is, I think, because I was still logged in under the everyday account.

When I logged out of the everyday account, and logged into my admin account (instead of just switching straight to the admin account, leaving my everyday account logged in), then the “Allow user to administer this computer” checkbox for my everyday account was enabled, and I could uncheck it.