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

• https://support.mozilla.com/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"

The website may be using autocomplete=off to prevent Firefox from saving 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

• Saved Password Editor: https://addons.mozilla.org/firefox/addon/saved-password-editor/

Thanks, cor-el. Additional details: On another site that doesn't have its own "save password" box I've tried the bookmarklet and Saved password editor 2.2.5. The latter adds the user name/password to my saved password list in Firefox tools but when I go to the site neither the user name, password nor "remember me" box appears, even if I type the first digit of the user name. My options are no private browsing, save passwords, no exceptions, accept cookies, no clearing of forms history.

Some sites like Windows Live (Hotmail) use JavaScript to create the input form and the Firefox Password Manager doesn't work in cases like that. You can use the extension to save the name and password in such a case, so you won't forget them, but Firefox won't fill them even if saved that way.