O'Day was born in Chicago, the son of railroad engineer James O'Day ( – 1885) and his wife Margaret ( – 1895), who were immigrants from Ireland and were both deaf. The couple had at least seven children: Daniel ( – 1898), James Jr. ( – before 1895), Catherine ( – 1901), Henry, Margaret ( – before 1895), Mary McNamara ( – 1924), and Joseph (1870–1885). The O'Days originally settled in Buffalo, New York; by the mid-1850s, they had moved to Cincinnati, and they relocated to Chicago around 1858. Henry was born in the vicinity of Ewing Street (later renamed Cabrini Street) and Jefferson Street, which was just one block north of the starting point of the Chicago Fire in 1871; however, by 1870, the family had moved about two miles west to 1022 W. Jackson Street (renumbered in 1909 as 2433 W. Jackson Boulevard), which remained the family residence until the early 20th century. O'Day worked as a steam fitter in Chicago before entering organized baseball.