Make sure that you not run Firefox in Private Browsing - Use Firefox without saving history mode. In Private Browsing mode some menu items are disabled (grayed) and some features like visited links and others are disabled and not working. You are in Private Browsing mode if you see "Tools > Stop Private Browsing".

See also http://kb.mozillazine.org/User_name_and_password_not_remembered

Thanks cor-el but that isn't the answer. Ive already checked out every Firefox setting & they're all OK. Any further ideas would really be appreciated

I assume that you have enabled the Password Manager? Tools > Options > Security: Passwords: "Remember passwords for sites" Do you see any passwords if you click the "Show Passwords" button? Are you using a Master Password?

See also http://kb.mozillazine.org/Password_Manager (Troubleshooting)

Already checked all of the above, but further investigation indicates that the drop down box offering to save passwords doesn't appear only on certain sites. It appears that if the homepage of a site doesn't have a "remember me" option at login, Firefox doesn't ask to remember the password. This never used to be the case.

Peter, this may be useless advice, but I've seen reports stating that reinstalling Firefox fixes this kind of issues. Please try it and confirm if it does it for you. I hope it does.