Is there a certain novel that best represents each New York City neighborhood? Probably not, but a certain set of novels, perhaps. It's something the NYPL is currently looking at with this map, which will be filled with even more red dots as additional suggestions come through via crowdsourcing.
Staten Island, Queens, and The Bronx are really underrepresented here, but there's also a lack of literary dots in Brooklyn (which only has 10 novels listed so far).
Their current list of "famous New York City-based novels according to the neighborhood in which they take place" features American Psycho by Bret Easton Ellis (Financial District), Ragtime by E.L. Doctorow (Flatiron), The Great Gatsby (Midtown), Catcher in the Rye (Central Park), The Age of Innocence by Edith Wharton (Upper East Side), Tropic of Capricorn by Henry Miller (Williamsburg), The Bonfire of the Vanities by Tom Wolfe (West Bronx/Fordham), and more.