What is the name of the large island that is part of New York?

If you're referring to New York State, then the answer is Long Island.

If you're referring to New York City, then I believe the answer you're looking for is also Long Island. However, the question would perhaps have been better worded as: "What is the name of the large island that part of New York City is situated on?"

At 1400 square miles (3626 sq km), Long Island is the largest island in the continental United States. The entire island is part of New York State. Two of New York City's five boroughs (Brooklyn and Queens, aka Kings County and Queens County) occupy the western end of Long Island. The remainder of the island consists of Nassau and Suffolk Counties, which are part of New York State but not New York City.

So the entire island is not part of New York City. Only the western end (the part that Brooklyn and Queens are situated on) is part of New York City.

The borough of Manhattan and the borough of Staten Island are the only two islands that are entirely within the boundaries of New York City (not counting all the small to miniscule islands dotted around the waterways that surround the city). But Manhattan Island and Staten Island can hardly be called "large" islands. Manhattan is only about 34 square miles (88 sq km) in total area, and Staten Island is a little over 100 square miles (259 sq km).

The remaining borough -- The Bronx -- is attached to the mainland United States. It is the only New York City borough that is not an island or part of an island.