Received wisdom in Ireland has long been that the process of reclaiming and resuming the Gaelic patronymic prefixes "Mc" (mac, "son of") and "Ó" ("grandson of") paralleled the Gaelic Revival in the second half of the nineteenth century, probably accelerating as the Revival grew in strength in the early decades of the twentieth century.
The figures from birth registrations tell a different story. The proportion of total births recording Mc (or Mac, or M') was 10.14% in 1865. In 1913, it was 10.48%. So there was an increase, but certainly nothing dramatic and accelerating only very slowly. It is tempting to surmise that the great flood of "Mc" resumption only took place when it became clear in the early 1920s where power would lie in the new Ireland.
Interestingly, the story is different for surnames starting "O". In 1865, 1.67% of total births used "O' ". By 1913, it was 3.2%. Perhaps the difference is that "O" surnames are found predominantly in Munster (and Donegal), traditionally nationalist regions, whereas "Mc" surnames are concentrated in Ulster, with a unionist majority.