One of my older iPod Touches (IPod Touch 8gig – 2nd Gen) suddenly decided that it could no longer access Gmail and kept asking for the Gmail password. Repeated attempts failed and changing the password did not help. Meanwhile my new iPod touch and desktop access continued to work. In the end, disabling 2-step verification solved the issued.
Bit by bit, new apps and features are causing the old iPod to be less and less functional.