You probably still have references to that account in Thunderbird (Address Book contacts or Calendar).

Don't think that's it. I did have two references in the Address Book (the calendar is blank) but deleting them didn't get rid of the issue.

What about Chat? That seems to be another place where a Google account could have been added…

Do you have any add-ons that could also make use of the account? Start Thunderbird in Troubleshoot Mode and see whether the problem happens there as well:

https://support.mozilla.org/kb/troubleshoot-mode-thunderbird

The problem window showed up about 15 seconds after I established Troubleshooting mode. Saw no sign of any addons (which is a relief, because I haven't added any).

I primarily used the offending account from a G-mail page in Chrome, because the association had an email list of around 250 members, and we wanted to hide the members mails from other members.

You didn't say whether you looked in Chat… If that's not either, I'm out of ideas…

I did have two references in the Address Book (the calendar is blank) but deleting them didn't get rid of the issue.

I have no idea what that means. IF you do have an address book which is linked to your Google Contacts, you need to delete the address book in Thunderbird. Just deleting the contacts within the address book won't cut it. The same basically goes for a calendar which is linked to your Google Calendar. You need to delete the calendar in Thunderbird.

Before I go through all that... it is a pain, after all. What's the mechanism that this cures?

I've deleted the association user profile from my computer, so that should take out the Google contacts for that user too. How is this broken-on-both-ends linkage still causing problems?

Before you go through all that… Make sure it is what you're being suggested to do, because I have the feeling you had already done what christ1 said, i.e. remove any references to that Google account from Thunderbird. The fact that you seem to think that would be a pain makes me believe you misunderstood that as meaning something else and I don't even understand what are you talking about now. This thread is becoming a bit messy… :)

The part I thought would be a pain was deleting the Thunderbird database. I was assuming that I couldn't just restore it from a current backup or the problem would return. I have eliminated references to the account in question from the address database, but I didn't wipe out the whole data base and restore it.

My question still stands, though. With the account in question eliminated from Thunderbird, and the address references removed from Thunderbird, what is happening to trigger a log-in reminder every half hour or so?

One other fact may be relevant. This occurs only on my desktop computer, and my desktop computer is the only place I accessed that account via a Chrome page.