Rasmus Finnestad, brother Ole and sisters Sarah and Mary all arrived in the United States in 1875. I had attempted to locate the 1880 census records for them under the name "Finnestad" but was unable to find anything. Then I decided to use "Oleson" and found Rasmus.
Below is the 1880 US Census for Dement Township in Ogle County, Illinois.
On line 34 is Rasmus Oleson, a hired hand for the Countryman family. Rasmus Oleson Finnestad was born November 20, 1863 in Norway, so his age at the date of the census on June 11, 1880 would match.
It's not too far-fetched to believe I am on the right track as by the time of the 1900 US Census, Rasmus and family were still living in Ogle County.
Growing up, I was told our ancestors used the name Oleson and then changed it to Finnestad after discovering there were so many Oleson (Olson, Olsen) families living in the area.
This also fits with the history of Norwegian naming patterns.
Another reason I believe this is Rasmus Finnestad is that his older sister, Iverdine used the last name of Oleson when she was married in 1872 to Lewis Travland.
I have run into other examples where the family had used "Oleson" rather than Finnestad, so I'm pretty confident this is the Rasmus Finnestad.