Barnafoss has been evolving through recorded history, as the river has dug itself down through the lava and runs in a deep and narrow ravine. There used to be a stone arch spanning the river, forming a link between the districts of Hįlsasveit and Hvķtįrsķša. The following story relates the reason for the name "Barnafoss" (Children's Falls):

Once there lived a widow on the farm Hraunįs. She was well off and among her possessions was the farm Noršurreykir in Hįlsasveit. She had two children. They were quite young when this story took place. There came a time when evening services were to be held at Christmas at Gilsbakki. The Mistress of Hraunsįs and all her folk, except the children, attended the service. The children were told to stay at home and play. The moon was shining and the weather was fine. When the people came home the children had disappeared. Their footsteps led to the stone arch over the river. Their mother had the arch destroyed, saying that no man would ever be allowed to cross the falls alive. (Kristleifur Žorsteinsson II (1972) Hvķtį.276)