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I think Nilson is "son of Nils". However, I know that a lot of people with the last name of Neilsen (son of Neil) change their name to Nilson or Nelson once they come to the States (in order to make it easier in terms of spelling and pronunciation). So, Nilson could be "son of Neil"
Still, if Nilson is your father-in-law's given name, it was probably meant to be Nilson and not Nielsen, so I guess it means "son of Nils".
Hope that helps