Bernard, my understanding is that the two spellings are variants of the same name. The Gaelic from which it is taken in Scotland is Mc or Mac Donnachaidh, and McC's have generally considered themselves as part of the Scottish Clan Donnachaidh. It is also known from the research, however, that those of the name were also affiliated with Clan Campbell. I asked Mr. Kelly who seems fluid in Irish Gaelic just days ago what the Irish Gaelic root is for the name, and I will quote his response: "Go raibh maith agat as do cheist, a Susan. / Thank you for your question, Susan. Ninsa, or Not difficult, as our ancestors would say. McConnaughey is usually a phonetic for Mac Dhonnchaidh, Son of Donnchadh, usually translated as Brown Warrior. Also, I should recommend Bell's Ulster Surnames, MacLysaght's Irish Surnames series, and Woulfe's Sloinnte Gaedheal Is Gall (Surnames of the Gaeil and the Gaill). These are excellent books in English which almost always provide good information on original names, locations, and more. Go raibh sé sin cabhrach. / Hope that's helpful."