The problem with determining the origin of surnames in the whole archipelago of the British Isles is the sheer number of linguistic influences that have affected these islands.
It is as well to try and identify the etymology of the name, ans that will also point to the place of origin. Although, as is discussed on here on many occasions, the firming up of surnames in the British Isles occurred at different times and may have altered over time as well.
My guess is that this name has no connection with any of the Celtic/Gaelic languages and therefore would be more likely to originate in England.
The surname database website claims that it is derived from a patronymic based on the word May, a pet form of the male name Matthew or its earlier Greco-Hebraic forms. (see www.surnamedb.com)
The idea that there was a family known as Mayes or May in the Kintyre area is not ruled out of course. There is a possibility that Mayes also locally derived from the Brythonic Maes. Kintyre was not far north of the old Kingdom of Strathclyde which was a Brythonic speaking area. I cannot speak about Clans as I have no idea about them, although my old boss told me that a vast number of people laid claim to membership of a clan without any rights or actual familial or kindred links. He was a clan chief, something that he was immensely proud of and at the time he was also Lord Chamberlain to HM The Queen, and a Knight of the Thistle.
He was clear that Sir Walter Scott had a lot to answer for in creating false histories in Scotland.