1630s, "fact of not depending on others or another, self-support and self-government;" see independent + -ence. Earlier in same sense was independency (1610s). U.S. Independence Day (July 4, commemorating events of 1776) is recorded under that name by 1791.
An Old English word for it was selfdom, with self + dom "law," but in form this is closer to privilege (n.). The two concepts are not always distinguishable.