Yes, Grover and Frances Cleveland had two children during President Cleveland’s second term (1893-1897), including the only child of a President actually born IN the White House — Esther Cleveland, born in 1893. Other children have been born in the White House itself, including grandchildren of President Jefferson, John Quincy Adams, President Van Buren, President Grant, and President Wilson, but Esther Cleveland was the only child of a President actually born in the building.
The only other First Family who had a baby during their Administration were the Kennedys but, like so many other events in the history of the Kennedys, it ended tragically. Jackie Kennedy gave birth to Patrick Bouvier Kennedy on August 7, 1963 (just three months before JFK was assassinated), but the baby was born about six weeks early and died two days later. Also, John F. Kennedy, Jr. was born about two-and-a-half weeks into the transition between JFK’s election in November 1960 and his Inauguration in January 1961.