Thomas Stanton is a common name not only in Ireland, but in the rest of the British Isles. The name Stanton derives from "stone town" of which there were many and hence there are many unrelated people who chose Stanton for a surname.
I see from ancestry.com that there are 2 Thomas Stantons in the U.S. born in Ireland in 1800. One was married to Bridget Dougherty, lived with his son Edward and family in Cleveland, OH in 1870. One was married to Margaretta and resided in New Haven, CT in 1850.
There was a Thomas Stanton, born 1800 in Ireland who had a son, Edward Stanton, born 1842. The U.S. Immigration records list this Edward to have arrived in Boston, U.S. from Ireland on 7 Feb 1871. This record does not list County Mayo.
There is another Edward Stanton, born 6 March 1846, who arrived Boston, U.S. 28 June 1861 from County Mayo Ireland according to U.S. Immigration records.
I believe that the Thomas you are referring to, who is my 3rd great grandfather, is the father of the Edward born in 1842 and is NOT from county Mayo, while the Edward born 1846 is. I base this on the immagration records and the 1870 census. (If this is MJH who wrote this, I'm JA.) Feel free to review my documentation and tree at ancestry.com. Username, Jarmbru, Heim & Roche Family Tree.