The last names of the Icelandic are created through the very specific system of taking the first name of the child's father and appending "són" or "dóttir" as appropriate.

To take the example of the only Icelandic person anyone here is likely to have heard of, Björk Guðmundsdóttir is the daughter of Guðmundur Gunnarsón ("Guðmund's Daughter"), and has a son named Sindri Thorsson ("Thor's son"). (Thor, if you're wondering, was the guitarist for the Sugarcubes.)

There are, of course, examples out there of people choosing to buck the system and retain the surnames of their parents rather than naming their surnames after their parents; i'd imagine if you looked you'd find someone somewhere who named their surname after their mother. However lining this kind of thing patriarchally is the default and apparently not thought about much.

The whole thing is, i think, somewhat nifty, and certainly makes things less confusing in terms of working with Genealogy. (Iceland has, as it happens, a long tradition of producing Genealogy freaks.)

As a result of this rather unique relatively unique system, Icelandic phone books list people alphabetically by given name.