Norwegian censuses of 1910, 1900,1875 (only partially online), 1865 and 1801 are online as searchable databases.Two websites with different search functions and strengths can be used. (http://www.rootsweb.ancestry.com/~norway/DigitalArchives.html) (http://www.rhd.uit.no/indexeng.html)
Tips on Using Digitalarkivet (http://www.rootsweb.ancestry.com/~norway/DigitalArchives.html)
Mollie is not a typical Norwegian name - the woman you are looking for is likely to have changed her given name in some manner.
A female born in 1865 Norway would be expected to have been identified with her patronymic name (her father's given name + datter), in Mollie's case that should have been Antonsdatter or Anthonsdatter.Very very few would have been identified with her father's patronymic name, 'Nelson'.
'Anthon' might also have been spelled as Anton.His patronymic name you spell as 'Nelson' was probably spelled as Nilsen or Nielsen in Norway.
You should try to find more identifying information about this woman and her family.When emigrated from Norway?Any clues given as to where in Norway they lived?