As an English name it can also derive from the month of August, which was named for the Roman emperor Augustus.
The most notable bearer of the name was Cyrus the Great, the founder of the Achaemenid Persian Empire in the 6th century BC. He is famous in the Old Testament for freeing the captive Jews and allowing them to return to Israel after his conquest of Babylon. As an English name, it first came into use among the Puritans after the Protestant Reformation.