From my UCLA colleague Kendra Willson’s Political Inflections: Grammar and the Icelandic Surname Debate, pp. 135-37:
Only some 15% of contemporary Icelanders bear surnames inherited in a fixed form. A person’s first name remains his or her primary name, while the indication of whose son or daughter he or she is is viewed by Icelanders less as a name than as a secondary descriptive label. The fact that the Icelandic telephone catalogue is organized by given name is a source of wonder to foreigners and a locus of national pride for many Icelanders….
[S]urnames entered Modern Icelandic usage [starting with the 17th century]…. Over the following two centuries, the assumption of surnames by members of the upper and upwardly mobile classes became more and more common….
The first official [but unsuccessful] attempt to stem the tide of surnames was a proposal presented to … the Icelandic parliament, in 1881…. This law would have required Icelanders to obtain royal permission before adopting a surname, as well as exacting a fee of 500 crowns … and an annual [tax] of 10 crowns per syllable of the last name.