Avatar dates to early graphical bulletin boards and MUDs in the 80s. Originally it was used as a synonym for "Player Character" (after Moorcock and his Avatars of the Eternal Champion) and spread from there. PFP is a post-facebook thing.
I wonder if the Ultima series had anything do with that, since the player character was called the Avatar. NATOpedia seems to support that, claiming that Ultima IV popularized the use of the term avatar in this context.