From JCHAS:

Barretts, from the Barretts, an ancient Anglo-Irish family, the owners fo the castles of Ballincollig, Castlemore, and Carrigrohane (or Kilgrohane). The head of this family styled himself "chief of his nacion." He was a feudatory of Muskery. It is said that O'Neill, Earl of Tyrone, when passing by Castlemore, near Mallow, to assist the Spaniards at Kinsale in 1600, inquired who lived in that castle, and being informed one Barrett, who was a good Catholic, and his family had possessd the estate for over four hundred years, exclaimed in Irish, "No matter, I hate the English churl as if he came but yesterday!" In 1600 William Barrett, of Ballincollig, submitted to the Queen's mercy, having been concerned in the Desmond rebellion.

The 1659 census did not enumerate by household. However, surname frequencies were noted. Below is a list of the surnames found in Barretts, in descending order of frequency.