The Norman arrival in Ireland in 1169 was just one end-point of their extraordinary expansion out of Flanders and northern France between the eleventh and the fourteenth centuries.
Superior military technology, deployed 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 yet using true hereditary surnames. The eldest-son-takes-all practice of feudal primogeniture meant younger sons had to go off and fend for themselves, one of the factors that drove their expansion. Perhaps that fracturing of Norman elite families weakened the need for hereditary names to signal wider family and tribal connections.
But the Gaelic Ireland they overran was in the middle of an explosion of hereditary 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, High King of Ireland and victor at the Battle of Clontarf, understandably wanted to flag up their connection, and adopted Ó Briain. But the sons of one of those grandchildren, Mathghamha Ua Briain, picked their own father as an origin point and became (in modern Irish) Mac Mathúna, McMahon, son of Mahon. 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 familiar modern names: Davey, Davitt, Doak, Galwey, Gibbons, McNicholas (Mc)Philbin, Gillick, Jennings, McRedmond. All stemmed from the forenames of prominent de Burgos, and all followed precisely the Gaelic Irish O and Mac tradition.
The upshot is that almost all so-called Norman surnames were created and adopted only in Ireland. “Hiberno-Norman” is little grudging. They are Irish surnames.