Marien, in a personal email you mention that the marriage of Phoebe and Noah was in Vigo County, Indiana in 1865. I find the Noah and Phoebe you mentioned in Bement, Piatt County, Illinois in 1870 as Noah and Phoebe Gillman. I can't find the marriage record online.
The age of 23 in the 1870 Census record matches that of Phoebe Ketchum, daughter of Eleazar Ketcham and Sarah McKee. That would explain the son's name.Young Eleazar Gilman is 1 year old, born in Illinois.
Eleazar Ketcham and his family are found in Illinois in 1870, in Coles County.
The marriage record for Noah Gillman (sic) and Nancy Scottis is in Coles County in late 1874. In 1880, we see young Eleazar with father and step-mother, plus apparently two full sibs and one half-sib, in Douglas County, Illinois per the Census. After that I have not researched.