Ok, so it's just "City" (like many NYers on the forum have stated, that's just what everyone calls NYC and have for YEARS and YEARS) and that's fine for a shortened version of the club's name. But how about a nickname for the club/fans along the lines of:
Real Madrid = Los Blancos, Bayern Munich = Die Bayern (The Bavarians), Juventus = the Old Lady, PSG = Les Parisiens (The Parisians), Manchester United = the Red Devils, Arsenal = the Gunners, Manchester City = the Citizens, Tottenham Hotspur = the Spurs, West Ham United = the Hammers, etc.
So far people have suggested: the Tokens, the Bluebirds, the Blues, the Blue ______ [insert anything here], the Amsterdamned, the Mankees, The [New] Yorkers, the Insomniacs, the Pentagons, the Yanks, the Concrete Citizens. I'm sure I have missed one or two (I've included as many as I could think of, even the ones that were suggested in a joking manner), so if I missed your favorite suggestion please reply so it's included. I don't mean to exclude any here, just trying to get a roundup of the suggested nicknames.
I just thought of another one as I was writing this post: the Emperors (as in the Empire State and the Evil Empire which seems to be how we are perceived by other teams/fans). Also, the Straphangers name has been offered by someone as an offshoot to the Third Rail (I cannot remember the precise suggestion, maybe a casual group of Third Railers?), but I think the Straphangers is such a good name that it could be used as the club's nickname.
There are a few that I think are pretty legitimate, including (but not limited to) the Tokens (for the subway token/NYCFC badge), the Straphangers, the Blues (although that is Chelsea's nickname, I still think it works because we are in MLS, not the EPL) and the Bluebirds (for the New York State bird: the Eastern Bluebird). Also, there is quite a lot of precedent for using birds: Newcastle United = the Magpies, Norwich City = the Canaries, Crystal Palace = the Eagles, Swansea City = the Swans. Many others that I wasn't able to think of as well, I'm sure.