His exact parents are unproven, but, his father was Samuel Davis and his mother was Mary (last name unknown). It is not proven when or where he was born. A lot of confusion, as he was christened in Ohio, but we know from Quaker records he lived with his parents in Pennsylvania. They were a part of the Radnor Meeting house.
On 13 Nov 1767 at the Radnor Monthly Meeting, Abiathar Davis is listed as living in Delaware, Pennsylvania His mother is shown as Mary Davis during his admittance to the church
On 24 Feb 1770 Abiathar Davis is listed with his father, Samuel Davis with a residence of Fairfax at the Radnor Monthly meeting as a "Removal" This is probably when he moved to Georgia.
He was a native of Pennsylvania and a Quaker, he moved first to Georgia, but being opposed to slavery, he ended up in Ohio. He was the father of 4 sons and 5 daughters. 
Biographical and Genealogical History of Cass, Miami, Howard and Tipton Counties, Indiana, Vol. 1, Chicago: The Lewis Publishing Company (1898), Page 88
Brogan, Hugh and Charles Mosley, American Presidential Families, New York: Macmillan Publishing Company (1993), Page 635.
McLean, Hulda Hoover, Genealogy of the Herbert Hoover Family, Stanford, California: The Hoover Institution on War, Revolution and Peace, Stanford University (1967), fold-out chart between pages 4 and 5.
Roberts, Gary Boyd, comp., Ancestors of American Presidents, Boston: New England Historic Genealogical Society (2009), Page 112.
Samuel Davis and Mary (Gorham) Davis were not his parents; there is a complete listing of their children cited in Samuel's biography and Abiathar is not listed. In addition, neither of them had ever been to Ohio or Pennsylvania that I know of.