have a look at this article:

Usernames and passwords are not saved

also, the following article may come handy:

Password manager - Remember, delete, change and import saved passwords in Firefox

Websites remembering you and automatically log you in is stored in a cookie.

• Create an allow Cookie Exception to keep such a cookie, especially for secure websites and if cookies expire when Firefox is closed.
• Tools > Options > Privacy > Cookies: Exceptions

In case you are using "Clear history when Firefox closes":

• do not clear the Cookies
• do not clear the Site Preferences

• Tools > Options > Privacy > Firefox will: "Use custom settings for history": [X] "Clear history when Firefox closes" > Settings
• https://support.mozilla.org/kb/Clear+Recent+History

Note that clearing "Site Preferences" clears all exceptions for cookies, images, pop-up windows, software installation, and passwords.

Clearing cookies will remove all specified (selected) cookies including cookies that have an allow exception and cookies from plugins.