Björgvin was born in Reykjavík, Iceland in 1977 to parents, actors Edda Björgvinsdóttir and Gísli Rúnar Jónsson.

In the late 2000s, Björgvin got the job as the children's presenter at Stundin okkar. Stundin okkar is one of the longest-running television programs in Icelandic television history. It has run at 6 PM on RÚV Sunday nights every year since Christmas 1966. In addition to hosting the show, Björgvin also published adventure books for children with themes related to the children's show. Björgvin wrote the first episodes with his mother Edda and said it was the most fun job in the world, but in 2011 he felt that it was his time to pass the torch. By that time Björgvin had won an Edda award for best children's television program.