On a trip to Ireland a few years ago, we asked that same question O'Cathail , etc. came out as "fierce warrior" according to a local; They teach it in school according to her. The name has morphed several times. There is a Catherine Cahaley in a grave at Cashel castle in Tiperrary. A bartender in Shannon dropped a white pages on my table filled with 2 pages of Cahaley's, all presumably related in one way or another. The point is, the name has changed over the years because many of our ancestors could not read or write (thanks, Brits) so finding the links are very difficult. I found a James Cahaley from Ireland with DNA roots from the 1800's, and that is very good. He was from Roscommon, as I recall, but linking fathers proved nothing. There are gaps because my 4th was here in America in 1760, and from there my quest ends for now.
Roscommon, Tipperary, even some in the north of Free Ireland pop up, but connecting them is difficult. I have James Cahalin, James Cahaley, William Cahele, John Cahale, William Kahaly (there are no "K" Surnames to be found in Ireland as I know it), Catherine, Catharine Cahaly, Joseph Cahala, etc, and somehow they are all our cousins. I'm 73 now, and have been searching for over 40 years off and on, maybe more. If you find something, let us know.