Were you using a password add-on?

Start Firefox in Safe Mode {web link} by holding down the <Shift> (Mac=Options) key, and then starting Firefox.

A small dialog should appear. Click Start In Safe Mode (not Refresh). Is the problem still there?

Type about:preferences#security<enter> in the address bar. Note: For Firefox 56+ its; about:preferences#privacy

Under Logins, Turn on Remember Logins. To check the passwords you have listed, press Saved Logins.

Warning: There are some web pages that do not allow browsers to record or fill in the log-in information. But if you go to the Mozilla Add-ons Web Page {web Link} (There’s a lot of good stuff here) you can find several add-ons that can override this, and give you better control of your password information. You can also try a Form Filling add-on.

A detail like a website remembering you (log you in automatically) is stored in a cookie.

You can use these steps to make a website recognize and remember you:

• you can create an "Allow" exception to keep specific cookies, especially in case of secure websites and when cookies expire when Firefox is closed.
• Options/Preferences -> Privacy & Security -> Cached Web Content: "Clear Now": Exceptions

Let the cookies expire when Firefox is closed to make them session cookies instead of using "Clear history when Firefox closes" to clear the cookies.

• Options/Preferences -> Privacy & Security -> "Use custom settings for history" -> Cookies: Keep until: "I close Firefox"

You can create an "Allow" exception to keep specific cookies.

In case you use "Clear history when Firefox closes" or otherwise clear history.

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

• Options/Preferences -> Privacy & Security -> Firefox will: "Use custom settings for history": [X] "Clear history when Firefox closes" -> Settings
• https://support.mozilla.org/en-US/kb/delete-browsing-search-download-history-firefox

• clearing "Site Preferences" clears exceptions for cookies, images, pop-up windows, and software installation and exception for password and other website specific data
• clearing "Cookies" will remove all selected cookies including cookies with an "Allow" exception you may want to keep