There is a John Cooper in the 1890 Veterans' Schedule who was living in New Brighton, Staten Island, New York. On the sheet, it says his rank was quartermaster, and under Regiment or Vessel, it says Brooklyn. It says he enlisted Sep. 1864 and was discharged Oct. 1866.
There is a Wikipedia page for John Laver Mather Cooper:
http://en.wikipedia.org/wiki/John_Laver_Mather_Cooper. His birthplace is given as Dublin, Ireland, which may throw a spanner/wrench into the works.
Sailors Snug Harbor, apparently the first home for "aged mariners" in the U.S., is still in business today as Snug Harbor Cultural Center and Botanical Garden, a National Historic District at the northwestern corner of Staten Island. Read about it at
http://en.wikipedia.org/wiki/Sailors_Snug_Harbor.
The John Cooper that Shelly refers to in the quote above may be the man you're seeking. West Brighton, Staten Island, where he was living in 1890, is the location of Snug Harbor. And "Brooklyn" under Regiment/Vessel may refer to the first USS Brooklyn, which was commissioned in 1858 and fought in the U.S. Civil War. To read about the USS Brooklyn, see
http://en.wikipedia.org/wiki/USS_Brooklyn_%281858%29John Cooper and other members of Brooklyn's crew are listed in the section of the Brooklyn article dealing with the Battle of Mobile Bay, one of the best-remembered naval battles of the Civil War, partly because Union admiral David Farragut yelled out "Damn the torpedos! Full speed ahead!"
The modern Snug Harbor's website is
http://www.snug-harbor.org/index.html.
Regards,
John