Dr. Tyrone Bowes, PHD best explains why so many tests don't match the surname expected or match many tests of differing surnames in his article "Using Y Chromosome DNA Testing to Pinpoint a Genetic Homeland in Ireland":

A son typically inherits two things from his father, his surname and his Y chromosome. The surname has changed considerably since his ancestor first adopted it, in Ireland it has been anglicized from its original Gaelic to English, often losing its Mac, or O’ in the process. Even its spelling in English has evolved over the centuries from, for example, O’Bouey, to Boe, and Bowe to its current form Bowes. The surname has often changed so much so that its original meaning in Gaelic can only be guessed at. However, in the estimated thousand years since an ancestor took his surname, the Y chromosome inherited from him remains virtually identical. This is assuming of course that he has inherited his Y chromosome, given that on average only 50% of individuals sharing a unique surname will have inherited the original Y chromosome of the founding ancestor. Where the Surname does not match the Y chromosome it is the result of what scientists refer to as a ‘non-paternal event,’ which encompasses such events as adoption, infidelity, and illegitimacy, often resulting in the maternal transmission of a surname.7 Only analysis of the Y chromosome will reveal whether maternal transmission has occurred.

Genetic genealogy for descendants of Irish ancestors is made easy for several reasons. Ireland was the first Country in Europe to adopt inherited paternal surnames. 478 These surnames were a genealogical record in themselves, denoted by Mac’ or O’ meaning son of, or grandson of respectively.

BigY700 DNA tests with associated autosomal (Family Finder - FF) tests are recommended.