The project I’m working on right now is a dungeonpunk fantasy novel set in a country that looks a lot like medieval Iceland and Norway and has a plot involving a young cutpurse who’s fostered by a dwarven fence. The narrator’s name is Bjarki Thórinsson. The patronymic is his foster father’s name given to him when he was adopted. He was born in a hospital (in the medieval sense) sent to a nurse and given a first name.
That leads to the question. What patronymic would Bjarki have been given before he was taken into the hospital? Is/was there a generic Icelandic patronymic for a child whose parents are unknown? The fact that he doesn’t have any known relatives is an important plot point.
I think in pre-Communist Russia, foundling boys were given the patronymic Abramov (son of Abraham). The same name is given to converts to Judaism – for their Hebrew name, they use Ben Avraham.
Maybe Iceland uses a similar name – Abrahamsson perhaps?
He likely wouldn’t have had a patronymic name at all. The use of patronymic names among common people didn’t become widespread until after the middle ages. I don’t know about the sagas but if you look at runestones you will see that people are only mentioned by their first name.