The way I see it, Niels Edvard Halvorsen apparently chose the name "Wilson" instead of "Halvorsen" when he immigrated to America. Peder has already posted the birth record for Kirstine and her marriage record. Here is the birth record for Hendrik:
Birth #45 (2nd from top)
Born 2 Jan 1877
Baptized 1 Jul 1877
Father: Nils Edvard Halvorsen
Mother: Kirstine Bernhardine Halvorsen
Kildeinformasjon: Telemark fylke, Solum i Solum, Klokkerbok nr. I 5 (1877-1881), Fødte og døpte 1877, side 5.
Permanent sidelenke: http://www.arkivverket.no/URN:kb_read?idx_kildeid=1414&i...
Note that the marriage record posted by Peder says that Kirstine was a widow when she married Nils Edvard Halvorsen. I would assume that "Stina" is a nickname for "Kirstine".
Kerstine's last name was not Olsen. Her father was Halvor Olsen, so she would have been called Kirstine Halvorsdatter.