https://www.re.is/blog/icelandic-names/ (found this not via the blessed Wikipedia, but by my favorite search engine 'kagi'.)
Traditionally, there are no family names in Iceland. Iceland is keen to hold onto the old patronymic (increasingly matronymic) custom whereby the child is given its fathers or mothers Christian name and is referred to as the son or daughter of. Iceland is a country that values tradition and customs and has a strong sense of family. But it also means theres no family lineage carried down through a single surname. You would imagine Icelands naming system would make it difficult to trace a family tree. Not so. The Book of Icelanders - Íslendingabók is used to trace genealogy and dates back well over a thousand years. In fact, its probably easier for Icelanders to trace their family tree than people from other countries. The extensive database, the relatively small population and a keen interest in family heritage - pretty much a national obsession - means the population keeps well up-to-date with their family lineage.