I'm Irish too, and in my mum's family the traditional naming pattern was used, give or take.
In the traditional Irish situation, neither mother nor father really had any input, and the naming pattern was followed in order to honour both the mother's and the father's lineage, so the two sets of ancestors would be honored (albeit with the man's parents first).
The names used were very commonly used in rural.Ireland of the 30s and 40s - for example, James, John, Patrick, Thomas, Bridget, Mary, Eileen, Theresa. It wasn't uncommon for more than one grandparent to have the same name - two Patricls or two Marys, for instance. In mum's family there were two grandfathers with the same name.
The pattern could also be interrupted by having all boys or all girls.
My other Irish grandparents didn't follow the traditional naming pattern at all, but all the names were saints' names, and they tended to be repeated randomly over the generations, with a few being dropped entirely after one airing.