Hapsy, Lydia, and Jimmy are three of Granny Weatherall's children. Lydia lives "eighty miles" away, and often drives that distance "for advice when one of the children (jump) the track". Jimmy also lives off on his own, but "still drop(s) in and talk(s) things over" now and then. The evening of Granny's death, Lydia and Jimmy drive over "as fast as (they can)", and arrive in time to speak with her briefly before she dies.
Hapsy is Granny's youngest daughter, and apparently her favorite. Granny describes her as the one "she really wanted", and in her confusion she "seem(s) to herself to be Hapsy also", mixing her own identity with that of the daughter she loves the best. Hapsy's whereabouts are not stated in the story, and although Granny asks for her several times as she lays dying, Hapsy does not arrive in time to see her mother before she dies.
Father Connolly is a priest who has come with "candle and a crucifix" to give Granny the last rites. Granny is Roman Catholic, as evidenced by her "pull(ing) out a rosary from the bosom of her dress, and her reference to the saints, a Mass, and receiving Holy Communion.