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KNUTSON Name Meaning and History
Respelling of any of the Scandinavian patronymics from the personal name Knut (see Knuth), for example Knudsen.
KNUDSEN Name Meaning and History
Danish, Norwegian, and German: patronymic from a personal name derived from Old Norse Knútr meaning ‘knot’ (Danish Knud, Norwegian Knut, German Knuth). This was a Danish royal name, Latinized as Canutus.
Sources: Dictionary of American Family Names, Oxford University Press, ISBN 0-19-508137-4, http://www.ancestry.com/facts/Knutson-family-history.ashx
About patronyms in Western Europe
In Western Europe patronyms were formerly widespread but later became confined to Scandinavia.
In Norse custom patronyms and matronyms were formed by using the ending -son (later -søn and -sen in Danish and Norwegian) to indicate "son of", and -dóttir (Icelandic -dóttir, Swedish and Norwegian -dotter, Danish and Norwegian -datter) for "daughter of". This name was generally used as a last name although a third name, a so-called byname based on location or personal characteristic was often added to differentiate people and could eventually develop into a kind of family name. Some Early Modern examples of the latter practice, where the patronymic was placed after the given name and was followed by the surname, are Norwegian Peder Claussøn Friis, the son of Nicolas Thorolfsen Friis (Claus in Claussøn being short for Nicolas) and Danish Thomas Hansen Kingo, the son of Hans Thomsen Kingo. Eventually, most Nordic countries replaced or complemented this system with the prevailing "international" standard of inherited family names. In Norway, for example, the parliament passed a family name act in 1923, citing the rising population and the need to avoid the confusion of new last names in every generation. The law does allow a person to retain a patronymic as a middle name in addition to the surname, as was common in Early Modern times; this is not a common practice, but does occur, a modern example being Audhild Gregoriusdotter Rotevatn.
In Iceland, patronymics are still used as last names and this is in fact compulsory by law, with a handful of exceptions.
In Finland, the use of patronyms instead of family names was very common well into the 19th century. Patronymics were composed similarly as in Swedish language or other Scandinavian languages: the father's name and the suffix -n for genitive plus the word poika for sons, tytär for daughters. For example Tuomas Abrahaminpoika (to be read in English as "Tuomas, Abraham's son") and Martta Heikintytär (to be read in English as "Martta, Heikki's daughter").
Sources: http://homepage.mac.com/sootica/Sites/DenmarkFAQ/namingtraditions.html, http://en.wikipedia.org/wiki/Patronymic