Thanks to an old book I picked up, Irish Families: Their Names, Arms & Origins by Edward MacLysaght (1957), I have the answer. (Cover of current edition at left.)
MacLysaght, at the time of publication, was the Chairman of the Irish Manuscripts Commission and also formerly Chief Herald of Ireland. MacLysaght's book is very well researched and indexed.
As I often do when I find a book on Irish surnames, I opened immediately to the index in the back, bracing myself for the fact that my Cowhey surname wouldn't be there. (I've had this experience many times.) But, surprise, there it was! One small page reference, but happily, Cowhey was mentioned in the index.
I turned to the page and found the following:
In Irish this name is Ó Cobhthaigh, pronounced O'Coffey as in English: it is probably derived from the word cobhthach, meaning "victorious". Coffey is one of those surnames which have not resumed the prefix O, dropped during the period of Gaelic submergence. Several distinct septs were prominent in mediaeval times, of which two are still well represented in their original homeland. These are O'Coffey of Corcalaoidhe in south-west Co. Cork, where local pronounciation often makes the name Cowhig or Cowhey, as in the place name Duncowhey, called after them. This sept is the same stock as that of the O'Driscolls...
So Ó Cobhthaigh it is! Below is the family's coat of arms, bearing a green background and three Irish cups.
After finding the Cowhey family in Irish Families, I found another source with similar information. According to The Surnames of Ireland, Cowhey is Gaelic for "descendant of Cobhthach (victorious)". Our particular Cowhey branch is probably a sept of Corcu Lóighdhe in west Cork related to the O’Driscolls. The family was seated at Dun Ui Chobhthaigh (Duncowhey) in the barony of Barryroe.
After so many years of wondering, it is a good feeling to have an indication of the area in Ireland where the Cowhey family probably originated: County Cork.
But as all new family history discoveries do, this one opens up a whole new set of questions. Where in County Cork did our Patrick Cowey come from? What records would have this information? I need more information before I can determine our own Patrick Cowey's connection to the county. And before I can plan that trip to Ireland...