I am far from being an expert but I think the most common thing to do is to just take on the surname of the parent as a sort of family name. So if Björk's father was John Smith then she would be Björk Smith. Family names are not unheard of and there are even a few Icelandic ones (although they are definitely the exception). Name laws in Iceland are really strict[0] but they mostly regard peoples "given name". I think you actually get to choose the surname of your child. I have seen cases where people have gone with the "-son" or "-dóttir" formula, typically in cases where the foreign name has an Icelandic counterpart, e.g. "John" and the Icelandic "Jón". The surname of the child could then be either "Johnson" or "Jónsson".
