There are five possible culprits on that brother, whose name is not hinted at. You can narrow it down a tiny bit by speculating that the brothers with wives and children wouldn't have time for a long bicycle expedition.
I found the following article in the Hay Springs (Nebraska) Leader, Friday, August 28, 1896
Alva Curtis Rash and daughter Pearl, of Gordon, Nebraska and his brother from Hardin County, Iowa were visiting with the families of J.S. Marshall (???) and Seth Francis Hammer (in Nebraska). Alva and Pearl returned home, while his brother continued his journey to the Black Hills (of South Dakota) on his bicycle, going through Hay Springs (Sheridan County, Nebraska) and Chadron (Dawes County, Nebraska). Mr. Rash will return via the bad lands (dry territory in South Dakota) passing through Pine Ridge agency (an Indian reservation in South Dakota) and Gordon to his brothers.
Brothers of Alva:
Columbus Elliott Rash, 1859-1943, married with 3 kids.
Benjamin Jesse Rash, 1866-1951. Single at this time, married in 1897.
Solon Lincoln Rash, 1869-1949, married with 3 kids
Henry Thomas Rash, 1872-1955, single at this time, married in 1902.
Andrew Seth Rash, 1877-1919. Unmarried, apparently, and very young.
