Name giving
Icelandic Naming Convention – Surnames
In Iceland, parents' names are used as surnames. This naming convention is based on children generally receiving their parents' names, either the father's, mother's, or both. Family names (surnames passed down through generations) are uncommon in Iceland and are only permitted with certain exceptions.
Therefore, family members in Iceland often have different surnames.
Examples:
Gunnar Jónsson and Anna Pétursdóttir have children named Sigurður and Elísabet. Their surnames could be:
Sigurður Gunnarsson, Elísabet Gunnarsdóttir.
Sigurður Önnuson, Elísabet Önnudóttir.
Sigurður Gunnarsson Önnuson, Elísabet Gunnarsdóttir Önnudóttir.
It is not permitted to use a hyphen or "og" (and) in the name.
Siblings could even have different surnames, if they wish, for example:
Elísabet Önnudóttir, Sigurður Gunnarsson Önnuson.
Endings of Icelandic Surnames
A new name based on a parent's name is created by adding the ending -son or -dóttir to the parent's name.
Boys receive the name of their father or mother, with the ending "-son."
For example, Jón Sigurðsson if the father is named Sigurður or Jón Guðrúnarson if the mother is named Guðrún.Girls receive the name of their father or mother, with the ending "-dóttir."
For example, Embla Sigurðardóttir if the father is named Sigurður or Embla Guðrúnardóttir if the mother is named Guðrún.
More endings for surnames are permitted, such as "-bur." An individual with a neutral gender registration in the National Register is permitted to use this ending after the father's or mother's name.
They are also permitted to use the parent's given name in the genitive case without an ending.
The word "bur" comes from Old Norse and refers to "offspring."
Examples:
Blær Jónsbur.
Frost Ragnheiðarbur.
Links of interest:
Icelandic Names Act (in Icelandic)
Registers Iceland
Information regarding names on the website of the Ministry of Justice (in Icelandic)

Service provider
Registers Iceland