The name Sweet Owen dates to John C. Breckinridge (Wikipedia). Breckinridge was a popular Kentucky politician who first came through Owen County campaigning for congressman from Kentucky's Eighth congressional district in 1851. Breckinridge won his election by 537 votes. He carried Owen by 677 votes.
But the deal was sealed two years later. Breckinridge won his election by 526 votes; he won Owen by 875. The total number of votes cast exceeded the number of registered voters by 123 votes. When hearing the news, Breckinridge is said to exclaimed "Sweet Owen!"
Shortly thereafter, Breckinridge and his friends began referring to Breckinridge's youngest son, John Witherspoon Breckinridge as "Owen County," later shortened to "Owen." The boy was called Owen for a significant portion of his life.
John C. Breckinridge was Buchanan's vice-president in 1856, and ran for president in 1860, but lost to a congressman from Illinois named Lincoln.
This is written from information found in Mariam Sidebottom Houchens' Sweet Owen: The History of Owen County.