I agree this is a confusing family to research.I am helping my sister-n-law with this line.I think I understand where you thought Sarah's name was HAMILTON...probably from the 1870 census.I'll get to that .
Sarah Ann FANIN was born June 1857 in Perry co. IL Agness Ann HAMILTON married Joseph M. FANIN on 15 Nov. 1855 in Perry co.
1860 Perry co. IL Twp 5, Range 3 W Joseph FANIN age 27 b. ILL Agnes A FANIN age 22 b. ILL Sarah A. FANIN age 3 b. ILL Samuel FANIN age 1 b. ILL
1870 Perry co. IL Tamaroa twp FANNIER, Agness A. age 32 b. IL HAMILTON, John age 35 b. IL HAMILTON, Sarah A. age 13 b. IL HAMILTON, Samuel age 11 b. IL HAMILTON, Ena M. age 9 HAMILTON, Susan age 7 b. IL
Agness Fannin married George AYRE on 22 Mar. 1871.
1880 Perry co. IL, Tamaroa twp AYRE, George age 59 b. SC AYRE, Ann age 43 b. IL wife AYRE, Joseph age 7 b. IL son OSBORN, Sarah age 23 b. IL Stepdaughter OSBORN, Etta age 3 b. IL granddaughter OSBORN, Ida age 4 months born January
I believe that the 1870 census information is a mistake on the part of the census taker. Note that Agness is listed 1st, in the position as head of household. I think John HAMILTON is a brother to Agness. Note that Sarah and Samuel were listed as FANINs in 1860 and then HAMILTONs in 1870.