Read a late 70s interview with Murray, conducted by Martin A. Grove; published in his book Beatle Madness, (Manor, 1978) for more on Murray and the Beatles.
Murray's mother Jean suggested that "Splish Splash I was taking a bath" would be a good first line for a song. Bobby Darin and Murray wrote the lyrics; Murray's mother wrote most of the music. The song was credited to "Bobby Darin and Jean Murray" to avoid any possible suggestion of payola if the song became a hit. (Story told at http://www.songfacts.com/detail.php?id=1820 and retold in slightly different versions elsewhere.)