Why do male last names in Icelandic end with -son instead of -sonur?

Answer

In Old Icelandic, when the word sonr formed the second part of a compound, i.e. Magnús-son, Sigurðs-son, the final -r (-ur) was dropped in nominative singular, and the same morphology is used in Modern Icelandic. E.g.:

• Nom. Magnús Sigurðsson (son(u)r)
• Acc. Magnús Sigurðsson (son)
• Dat. Magnúsi Sigurðssyni (syni)
• Gen.Magnúsar Sigurðssonar (sonar)

In the word dóttir on the other hand, no change of this sort occurred. In Old Icelandic the ending was -er (-ir) in nominative singular and -or (-ur) for all other cases. Again, the morphology remains unchanged in Modern Icelandic.

• Nom. Guðrún Sigurðardóttir (dóttir)
• Acc. Guðrúnu Sigurðardóttur (dóttur)
• Dat. Guðrúnu Sigurðardóttur (dóttur)
• Gen. Guðrúnar Sigurðardóttur (dóttur)

Further answers in English:

• Are portmanteau words frequent in Icelandic? by Guðrún Kvaran
• Is Icelandic the oldest language in Europe? How old is it? by Jón Axel Harðarson
• What is the origin of the Icelandic language? by Guðrún Kvaran
• Are there rules that say when words in Icelandic should be masculine, feminine and neuter? For a foreigner it is not enough to add an article. by Guðrún Kvaran
• How did the Icelandic language start? by Guðrún Kvaran
• What is the shortest sentence in Icelandic to contain all the letters of the Icelandic alphabet? by Guðrún Kvaran
• How many words are there in Icelandic? by Guðrún Kvaran
• How many words are there in Icelandic for the devil? by Guðrún Kvaran

The original question was:

Why is it that male last names in Icelandic are in accusative -son instead of nominative -sonur, even when referred to in nominative, whereas female last names are referred to in nominative case -dóttir and not -dóttur?