Is this about making Firefox fill the name and password or about websites remembering you and log you in automatically?

Details like websites remembering you (log you in automatically) are stored in a cookie.

• Create a cookie 'allow' exception to keep such cookies, especially in case of secure websites and when cookies expire when Firefox is closed.
• Tools > Options > Privacy > "Use custom settings for history" > Cookies: Exceptions

In case you clear history, either manually or when you close Firefox:

• 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/remove-recent-browsing-search-and-download-history

Clearing "Site Preferences" clears all exceptions for cookies, images, pop-up windows, software installation, passwords, and other website specific data.

Clearing cookies will remove all specified (selected) cookies including cookies with an allow exception that you would like to keep.

It is about the password manager-saver. It no longer signs me in to sites that are saved or offers to save passwords. I can open the manager with my master password and see the sites and saved passwords. The proper boxes to allow it to sign in and save passwords are properly checked.

Do you get the drop-down list if you (double) click the input field or start typing a saved name?

Make sure that you do not run Firefox in permanent Private Browsing mode (Never remember history).

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

• Tools > Options > Privacy > Firefox will: "Use custom settings for history"
• Deselect: [ ] "Always use Private Browsing mode"

Some web sites send a signal that tells browsers not to auto fill passwords. I have found, however, that form filling add-ons are not effected by this.

Go to the Mozilla Add-ons Web Page {web link} (There’s a lot of good stuff here) and search for what you want.