score:7
It is likely that the establishment first offering pizza in NYC is now lost in the mists of time.
Having said that, you can check out Lombardi's Pizza, 22 Spring Street (at the corner of Spring and Mott; also a one-time boundary of New York's Little Italy). Lombardi's has been operating on the same location since 1905, and for a pizza house, that's truly amazing. 109 years!
Take the E down town and alight at Spring Street. A if it's late.
While getting your Nathan's hot dog, you may be able to root around Coney Island and find something with similar roots. That would be the second logical choice after what's now left of Little Italy.
The other contender, and arguably the best known, is Rays Pizza. To my mind Ray's is the most tasty and I will invariably have one each time I'm in Manhattan. But Ray's claim to being the 'Original' is somewhat tainted as it began in 1964, a relative infant in the Big Apple's pizza scene.
Lombardi's is your best bet, with a recommendation to have a Siciliana style at Ray's.