Firefox will only remember a password if you click the Remember button on the sliding bar at the top that opens when Firefox detects a password.
If you mean a check box on a web page to remember you then such a choice is stored in a cookie on your computer.

• Websites remembering you and automatically log you in is stored in a cookie.
• Create an allow cookie exception (Tools > Options > Privacy > Cookies: Exceptions) to keep such a cookie, especially for secure websites and if cookies expire when Firefox is closed.
• In Private Browsing - Use Firefox without saving history mode all cookies are session cookies that expire if that session is ended, so websites won't remember you.
• Do not use Delete browsing, search and download history on Firefox to clear the "Cookies" and the "Site Preferences"

Clearing "Site Preferences" clears all cookies, images, pop-up windows, software installation, and password exceptions.

See:

• Cookies - Information that websites store on your computer
• Enhanced Tracking Protection in Firefox for desktop
• Password manager - Remember, delete and change saved passwords in Firefox

On any site, FF 1a (and 10 before 11) ask(ed) me 'Do you want to remember the password "..." on "... ..."?' and I click the "Remember" button, but nothing happens (the button remains as is) and in fact the password has not been remembered (doesn't appear on the saved passwords list). I have FF set to keep passwords and form autocomplete data after shutdown, but neither passwords nor form data are saved (passwords already on the list do work). I use the NoScript addon, but this problem predates my starting to use that addon. Any idea what might be causing this problem and how to fix it? TIA.