'Gaelic' Variants
The Spelling Variants above mentioned the transformation of the Gaelic name MacGiolla Bríd to Gilbride and Kilbride. While the majority of Irish names can be said to have lost their O' or Mac (or Mc) stems, some family's always retained theirs. Thus while Reilly is commonly found surname, so to is O'Reilly.
Wider social movements, however, served to complicate matters. From around 1880 onwards a 'Gaelic Revival' movement gathered momentum in Ireland. This aimed to restore a connection with Ireland's past traditions. This saw a revision of interest in the Gaelic language, and in Irish customs in general. As a consequence by the early 1900s, some families that had, for generations been known simply as "Reilly" (to stick with this example) started using O'Reilly as their surname in all their dealings.
Irish Independence in 1921 intensified, if anything, this trend. Families that had been named for generations as Cartan became MacCartan, and Sullivans became O'Sullivans.