You probably have already confirmed that the feature is turned on here:

orange Firefox button or classic Tools menu > Options > Security > "Remember passwords for sites"

If you click the "Saved Passwords" button, do you see your login there? If so, try deleting it so Firefox saves it again. Perhaps it has become corrupted in some manner.

If you have an add-on that also saves passwords or forms data, there might be a conflict. You could test in Firefox's Safe Mode, which disables your extensions and ignores certain custom settings.

First, I recommend backing up your Firefox settings in case something goes wrong. See Back up and restore information in Firefox profiles. (You can copy your entire Firefox profile folder somewhere outside of the Mozilla folder.)

Next, restart Firefox in Firefox's Safe Mode (Diagnose Firefox issues using Troubleshoot Mode) using

Help > Restart with Add-ons Disabled

In the dialog, click "Start in Safe Mode."

If the login works correctly, this points to one of your extensions or custom settings as the problem.

Any change?

Do you still see the currently saved passwords in the Password Manager?

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

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

Make sure that you do not run Firefox in permanent Private Browsing mode.

• https://support.mozilla.org/kb/Private+Browsing

To see all History and Cookie settings, choose:

• Tools > Options > Privacy, choose the setting Firefox will: Use custom settings for history
• Deselect: [ ] "Always use private browsing mode"

Note that websites 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