Árni was born in 1663 at Kvennabrekka in Dalasýsla, in western Iceland, where his father Magnús Jónsson was the minister (and later prosecutor and sheriff). His mother was Guðrún Ketilsdóttir, daughter of archdeacon Ketill Jörundarson of Hvammur. He was raised by his grandparents and uncle. At 17 he entered the Cathedral School in Skálholt, then three years later, in 1683, went to Denmark (with his father, who was part of a trade lobbying contingent) to study at the University of Copenhagen. There he earned the degree of "attestus theologiæ" after two years, and also became an assistant to the Royal Antiquarian, Thomas Bartholin, helping him prepare his "Antiquitates danicæ" and transcribing, translating, and annotating thousands of pages of Icelandic material.