Rockefeller (1953)

Nelson and Winthrop Rockefeller were grandsons of oil tycoon John D. Rockefeller.
In 1953, Republican President Dwight D. Eisenhower appointed Nelson as chair of the President's Advisory Committee on Government Organization. He served as Governor of New York from 1959 to 1973. He was the 41st Vice President of the United States of America from December 19, 1974 to January 20, 1977.
Also in 1953, Winthrop Rockefeller--who was known as a playboy and hard drinker--moved from Florida and New York to Arkansas. It was jokingly said he moved there because he loved playing the banjo. Winthrop became Governor of Arkansas in 1966 and was said to be a great influence on future Arkansas Governor and U.S. President Bill Clinton.
Winthrop was probably the Rockefeller that Billy Joel was referring to, since his playboy antics were more in the news than things that Nelson was doing.
Nelson and Winthrop Rockefeller were the only brothers in U.S. history to serve as Governor at the same time.