Log in again after turning on the checkbox to stay signed in - and see if Firefox remembers this time. If you still cannot stay signed into Gmail, go back to the Firefox Privacy settings, click the Show Cookies button and delete the cookies the browser may have stored for Gmail and other Google sites. (These settings are under the Options icon in the Windows version of Firefox.) On the Preferences screen, click the Privacy icon on the left side to see your cookie settings if you do not see separate settings for cookies and browsing history, use the drop-down menu next to History and select “Use custom settings for history.” If Firefox is configured to delete all your cookies after you close the browser, your login preferences are being dumped as well. To get to your cookie controls, click the three-lined Menu icon on the right side of the Firefox window and select Preferences. While browser cookies - little strings of text - can be used by advertisers to quietly identify you when you visit certain sites, they can also store login information for websites like Gmail. If you are booted out of your Gmail account each time you quit and reopen the browser, make sure you have cookies enabled in your Firefox settings. I keep checking the box to stay signed into my Gmail account when I log into the Firefox browser on my Mac, but when I quit the program, the browser never retains the information and I have to keep logging in and typing in the security code that gets texted to my phone.

Stay signed in to gmail