Anderson is "son of Ander," with Ander the familiar short form of Andrew. English, Scottish, and Scandinavian naming all produced the same idea: Andersen in Denmark and Norway, Andersson in Sweden, Anderson as the usual English spelling.
Andrew itself comes from a Greek root meaning "manly," but families need the surname gloss, not a theology lesson. In the US the name is often Scottish or Scandinavian by ancestry, yet the type is always parent's name—parallel to Johnson, not to a trade like Miller.