Thunderbird doesn't even get to the point to prompt you for a password. There can be multiple reasons for this. See https://support.mozilla.org/kb/cannot-receive-messages for some troubleshooting steps.

What was the reason to delete the saved password in the first place?

I deleted the saved password because some advice found on Google—inluding the link above—said that is the way to set a changed password. (And because edit password was behind right-click, where I did not find it.)

JanEgil said

I was advised to delete the saved password to be prompted for the new one. (Better advice might have been right-click and edit?)

I would advise removal. Edit is there, but I just do not trust it not to end up with a trailing blank or something else silly.

But all Thunderbird says, is "styrheim@olivant.fo: Checking mail server capabilities...". For a very long time.

That could be because the anti virus is supplying the password and the user name.

It could also be because your account setting do not specify to use a user name. oAuth perhaps or even a setting of none.

I am no Wireshark expert, but it seems to me that Thunderbird never tells the server about the user name "styrheim@olivant.fo" ?

My normal response would be to disable any internet security suite including it's anti virus and firewall and see what happens next. On windows 10 that would be by starting the device in safe mode with networking. Microsoft offers the following instructions https://support.microsoft.com/en-au/help/12376/windows-10-start-your-pc-in-safe-mode

How can I get Thunderbird to ask for the new password? Or—can I somehow enter a password manually into the list? (I do have a Wireshark file, but it does not seem I can upload even a screenshot.)

Try the suggestion I have offered and see how it goes.

Sorry. safe mode made no difference.

I tried to configure a brand new e-mail account on the same server in Thunderbird. That also failed. So I guess there is some mismatch in the communication between Thunderbird and imap.olivant.fo.

I have contacted the mail server support team.

I found the culprit: An Android-phone with out-of-date password for another email on the same mail server: Caused out IP to be blocked. (Stupid clients keep trying the same invalid password over and over, and stupid servers counting repeated attempts with the same password as hacking.)

So my problem—at least concerning Thunderbird—is solved.

But I do not think that deleting saved passwords is good advice: If you have the new password, better rightclick and edit.

JanEgil said

If you have the new password, better rightclick and edit.

and then you go to get mail and have to fiddle with the drop down to get mail for only that account to test your new password. I have 20 email accounts in Thunderbird. I have no desire to wait for all of them to be polled just to check the new password works.. If it does not work back to the password manager and repeat.

Or you delete it and when prompted provide it. If it does not work it pops up again asking for the password. The only repetition is entering the use password manager check box over and over. Even if you did not delete the password (I never do) you still get the same dialog. Failure of it to appear means you have more issues than a password.