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It turns out the beach was named for Thomas Jones, an English privateer in the 16th century. He was granted power by the crown to attack ships that were enemies of England and take their cargo. In other words, he was a sort of legal pirate. It is unclear on the sources I found if he ever actively served as a privateer, but he was certainly a wealthy man by the time he made his was to North America. He eventually settled in Long Island when New York was still a colony and he was granted a large area of land that included Jones Beach. Also, he was granted whaling rights on the north and south shore of Long Island--quite a treasure. If you do a google search for Thomas Jones Privateer you can find a more information--some of it likely apocryphal.
He died in the early 1700's and was buried near Massapequa Creek. Interestingly, his family sided with the English in the Revolutionary War and the some of the Jones property was confiscated after the war and most of the remaining family went back to England. However, some Jones remained and held on to some of the property.
As an aside, I ran into this interesting document that has some before and after pictures from Jones Beach (photos shortly after it opened and photos from today). The pictures start on page 22. I thought my Long Island readers would enjoy them.