The Norman arrival in Ireland in 1169 was just one end-point of their extraordinary expansion out of Flanders and northern France between the 11th and the 14th centuries.
Superior military technology, used with ruthless brutality, allowed them to conquer and settle a vast swathe of the medieval world, from Byzantium in the east through parts of Britain, Germany, Italy and Spain, as far west as the Canary Islands. When they got to Ireland, they were not using true hereditary surnames. The eldest-son-takes-all practice of primogeniture meant younger sons had to go off and fend for themselves, a factor driving their expansion. Perhaps that weakened the need for hereditary names that signalled wider family connections.
But the Gaelic Ireland they overran was in the middle of an explosion of surname-creation, with great networks of extended family names budding and sub-budding off central stems as families grew or waned in importance. The grandchildren of Brian Ború understandably wanted to flag their connection (Ó Briain), but the sons of one of them, Mathghamha Ua Briain, picked their own father as an origin point and became (in modern Irish) Mac Mathúna, McMahon. Four generations later, Constantine (Consaidín) O’Brien, bishop of Killaloe, was the source of the Mac Consaidín line, the Considines. A great multi-generational flowering of names was taking place.
As they did wherever they settled, the Normans eventually integrated. They out-Irished the Irish when it came to fissiparous surname adoption. Just a single family, the de Burgos of Connacht, spun off dozens of modern names: Davey, Davitt, Doak, Galwey, Gibbons, McNicholas (Mc)Philbin, Gillick, Jennings, McRedmond … all stemming from the forenames of prominent de Burgos, all following precisely the Gaelic Irish tradition.
The upshot is that almost all so-called Norman surnames were created and adopted only in Ireland. “Hiberno-Norman” is too grudging. “Irish” will do.
The best popular account of Norman surnames in Ireland is by Dr Paul McCotter, available online at goo.gl/YMdDBg