"Remember me" details are stored in cookies. There may be a problem with the Google cookie, try deleting all Google/Gmail cookies then log in again and see if it remembers you. For details on how to do that see https://support.mozilla.com/kb/Deleting+cookies

Evidently, Mozilla has changed where they store cookies and how they store them. They used to be kept in a text file (signons*.txt) (the * is either nothing, 2 or 3) now they are kept in signons.qulite.

Go to Tools - Options - Privacy - if Permanent Private Browsing Mode is on, uncheck it in History, change the drop down to "Use Custom Settings for History" check Remember search & form history

close down FF then reload. Log on to gmail and click on "keep me signed on"

then close down gmail.

Re-open and it should have remembered your password. it did mine.