You can check if there are entries in the password manager with an empty user name.

Remove saved Password(s):

• Tools > Options > Security: Passwords: "Saved Passwords" > "Show Passwords"

You can also check the exceptions.

• Tools > Options > Security: Passwords: Exceptions

You can remove all stored data from a specific domain via "Forget About This Site" in the right-click context menu of an history entry (Show All History or History sidebar) or via the about:permissions page.

Using "Forget About This Site" will remove everything like bookmarks, cookies, passwords, cache, history, and exceptions from that domain, so be cautious and if you have a password or other data from that domain that you do not want to lose then make a note of those passwords and bookmarks.
You can't recover from that "forget" unless you have a backup of the affected files.

It doesn't have any lasting effect, so if you revisit such a 'forgotten' website then data from that website will be saved once again.

Thank you for your reply. As stated above I had already tried the first option, and the site isn't on the exceptions list either. I just tried the second option ("forgetting" the site) and that didn't work either. (I don't know whether it makes a difference or not, but I tried the "forgetting" option twice, re-testing the password both before and after closing the browser. The password prompt still didn't display.) Do you have any other suggestions?

The website may be using autocomplete=off to prevent Firefox from saving and filling the name and password.

You can remove autocomplete=off with a bookmarklet to make Firefox save the name and password.

• http://kb.mozillazine.org/User_name_and_password_not_remembered

If you can post a link to that site then we can check this.