Hello Mr. Mills,
My best suggestion is that you begin your search on your side of the ocean first. Working backward from virtually nothing is not only difficult, it is impossible.
Your ideal start would be to find Coakley Benjamin Mayo's record of birth. It will include not only his father's name, but also his mother's and may include her maiden name, and possibly their town or county of origin in Ireland. Unfortunately, Arkansas is one of those states which hold few records of birth pre-1914, with the exception of a couple of towns, but it never hurts to ask.
Barring that, see if you can find his record of death; it may supply you with some information. Do you know where he is buried? If so, go to the cemetery, view any details on the stone, ask if they have a cemetery register, and see if there is any information in it.
If you cannot find his birth record, you might want to look at the Arkansas Census records 1860, 1870, and so on, to see if you can locate his family.
Good luck to you with your research.