This likely denotes the domain of the "king of Uí Briúin", a title borne primarily by men of the Uí Briúin Seóla. As Knox points out, these kings were distinguished at an early date from the Síol Muireadaigh of central Roscommon in the Book of Rights, suggesting that the lands of the Uí Briúin Seóla were the original "Hy Briuin".
This probably refers to the domain of the "king of Uí Briúin", a title typically held by men from the Uí Briúin Seóla lineage. As Knox notes, these kings were distinguished early on from the Síol Muireadaigh in central Roscommon as per the Book of Rights, indicating that the lands of the Uí Briún Seóla were originally known as "Hy Briuin".