ugh, this is so annoying! i dealt with this too and it drove me nuts. for me, it was because i had "private browsing" turned on in my browser. if you're using incognito mode or something similar, facebook won't save your login.
try logging in normally (not incognito) and see if it sticks. if that doesn't work, maybe try a different browser?
try logging in normally (not incognito) and see if it sticks. if that doesn't work, maybe try a different browser?