As far as name "changes" (patronymic vs. farm name):
I've found that most Norwegians living in the US used their patronymic name (Halvorson, Anderson, Olsen, etc.) for the 1850-1880 censuses. In Minnesota, my ancestors began using their farm name when enumerated in the 1885 MN State Census.
Here is how your Ole Halvorson was enumerated in the US/MN censuses (as indexed by ancestry.com):
1870 Kenyon Twp: Ike[!] Halverson, born about 1834. Actual image looks like "Ole." With wife "Julia" and 4 children.
1875 Kenyon Twp: Ole Halverson, age 42, Aslaug, age 32 and 6 children and 3 "boarders."
1880 Kenyon Twp: Ole H. Verdal, wife Aslaug, and 6 children (indexed under "H. Ole Verdal.").
1885 Kenyon Twp: Ole Verdal, wife "Aslag" with 8 children.
1895 Kenyon Twp: Ole Halvorson, age 62 [widower]; living in MN for 37 years; enumeration district for 28 years; with children Halvor, John A., Martin, Jørgen, Thea and Karen. Also living with them is Randi Werdahl, age 21, born Norway and Gustoff Werdahl, age 70, born in Norway.
1900 Kenyon Twp: Ole H. Vardal, born April 1833 Norway, immigrated 1858(?) with 5 children.
1905 Kenyon Twp: Ole is not indexed (because the page is hard to read). But I think I found him, age 72, living in Kenyon Twp with George A. Werdahl/Verdahl [indexed under George A Harda????], age 44, born Norway.
1910 Kenyon Twp: Ole Halvarson, age 77, living with his daughter and son-in-law:
Name: Anne H Pederson [daughter of Ole]
Age in 1910: 41
Birth Year: 1869
Home in 1910: Kenyon, Goodhue, Minnesota
Relation to Head of House: Wife
Marital Status: Married
Spouse's Name: Aanon Pederson
Father's Name: Ole Halvarson
Father's Birthplace: Norway
Mother's Birthplace: Norway
Aanon Pederson 49
Anne H Pederson 41
Bertha P Pederson 19
Hannah T Pederson 17
Peder O Pederson 14
Alma O Pederson 9
Inga C Pederson 6
Alfred A Pederson 3
Ole Halvarson 77, born Norway
If you don't subscribe to ancestry.com, you should be able to find some of the above census records at www.familysearch.com
1877 Goodhue County plat map, Kenyon Twp: Ole Halvorson Wardal, owner of land in section 3 (just east of the Village of Kenyon).
1894 Goodhue County plat map, Kenyon Twp: O. H. Verdal, owner of land in section 3.
I'm sure you have this. Ole's MN Death Certificate:
WERDAHL, OLE H.
Date of Birth: na
Place of Birth: na
Mother Maiden Name: na
Date of Death: 09 Mar 1917
County of Death: GOODHUE
And a picture of his gravestone at www.findagrave
* All of my Norwegian ancestors settled in Goodhue County; I have access to many Goodhue County records. :)