I'm not sure this is the same Julia Halverson, but it seems to fit. I have a website for the 27th Iowa Volunteer Infantry (Civil War). I have a Paul Hulverson in Company I. I have been researching the men of the 27th. Research on Paul Hulverson showed his name to be spelled Halverson and Holverson. His tombstone has Halverson. Paul Halverson was married to Julia and had one son Halver P. Halverson. They lived in Allamakee County, Iowa. Paul Halverson died on March 25, 1900 and is buried in Oakland Main Cemetery, Makee Township, Allamakee County, Iowa, On the 1900 census, Julia was living with her son Halver and his family in Allamakee County. Then in 1905 they were listed in Beltrami County, Minnesota. In 1920, she was living with her widowed daughter in law (again in Beltrami County, Minnesota.) After that I cannot find her on the census.
On the 1860 census in Allamakee County is Larson Olson (age 56, born Norway), Julia Olson (age 50, born Norway), Celia Olson (age 12, born Norway), Ole Olson (age 8, born Iowa), Julia Halverson (age 23, born Norway) and Halver Halverson (age 1, born Iowa.) Paul was not on this particuar census, but their son was already a year old. Unfortunately the 1860 census does not show relationships, but these names are similar to the names you mentioned. Are you sure they did not also use the name Olson? I do note in looking at the census index, that someone has put REMA in their name.
It appears to me that we could be researching the same family. What do you think?