I think that the James W. Gibson, father of James W. Gibson, born 1871 would not be a fit as the 1880 Census shows them both being born in IL.
There is an Ancestral File showing James W. Gibson born 1826 Dickson, TN and died in Hamilton, IL 1907. He had a son, also named James, but the entry shows his middle name to be Henry, born 1855. James W. Gibson married Purdence Irene McGinnis 10 Dec 1850, who died in 1862. I'm guessing that she would be the mother of James born 1855.