Is this about passwords that you have saved in the Password Manager or is this about a website remembering you and log you in automatically when you visit this website?

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 -> "Use custom settings for history" -> Cookies: Exceptions

In the event that the passwords information is lost, you should record all information in a separate text file somewhere else on your hard drive, or written down. You can easily copy and paste if you need to do so. If you are concerned about someone else looking at that file, you can compress it using a password.

These can't get your data back, but will help in the future.

These add-ons can be a great help by backing up and restoring Firefox

https://addons.mozilla.org/en-US/firefox/addon/febe/ FEBE (Firefox Environment Backup Extension)

FEBE allows you to quickly and easily backup your Firefox extensions, history, passwords, and more. In fact, it goes beyond just backing up -- It will actually rebuild your saved files individually into installable .xpi files. It will also make backups of files that you choose.

https://addons.mozilla.org/en-US/firefox/addon/opie/ OPIE

Import/Export extension preferences