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

• create a cookie allow exception with the proper protocol (https:// or http://) to make a website remember you

You can check that you aren't clearing important cookies.

• using "Delete cookies and site data when Firefox is closed" keeps cookies with an allow exception
• using "Clear history when Firefox closes" to clear cookies clears all cookies including cookies with an allow exception you may want to keep
• clearing "Site Preferences" clears exceptions for cookies, images, pop-up windows, and software installation and exceptions for passwords and other website specific data

• Settings -> Privacy & Security
  Cookies and Site Data: "Manage Exceptions/Permissions"

These are linked together because sites could use local storage (also called DOM storage or offline storage) in your browser to work around cookie restrictions.

If there is one particular sites for which you need to preserve local storage content, you could create an exception -- the button is to the right of the setting:

That will also preserve its cookies.