I've looked at the census information, and think I'm on the family you are looking for.First, I used the LDS index for 1880.They had "Bunnino" listed, but it looks like "Runnion" to me.
1880 US Census - Ohio Grove, Mercer County, Illinois NA Series: T9Roll: 236Page: 398D Runnion W. MOORE, Self M W 36 Married Farm Laborer IN KY VA Susan M. MOORE, Wife F W 33 Married Keeping House NY NY NY Carra S. MOORE, Daughter F W 11 Single IL IN NY Martha MOORE, Daughter F W 9 Single MO IN NY Thomas C. MOORE, Son M W 6 Single IL IN NY Mary G. MOORE, Daughter F W 4 IL IN NY Bertha M. MOORE, Daughter F W 1 Single IL IN NY
I noticed that the second daughter was born in Missouri, so I looked there in the 1870 census, after not finding any reasonable matches in Warren or Mercer County in 1870.Here's what I found -
1870 US Census - Town. 45 Range 32 (Harrisonville PO), Cass County, Missouri NA Series: M593Roll: 767Page: 733Sheet: 9Date: 20 September 1870 Household 72, Family 72 Samuel Moore, 53 M W ____ $6000 $1000 Kentucky Martha W. Moore, 48 F W Keeping House Virginia Daniel Moore, 25 M W At Home "Indianna" Nannie Moore, 22 F W School Indiana James P. Moore, 20 M W School Indiana William W. Moore, 18 M W School Indiana Household 73, Family 73 Bromie W. Moore, 27 M W Farmer --- $200 Indiana Susan Moore, 23 F W Keeping House Indiana (Dittoed from above) Carrie Moore, 1 F W _____ Illinois
Now, back to 1880 for Samuel -
1880 US Census - West Peculiar, Cass County, Missouri NA Series: T9Roll: 679Page: 96A Samuel MOORE, Self M W 62 Wd Farmer KY KY IRE Milton PAINTER, SonL M W 38 Married Farmer KY KY KY Nannie PAINTER, Daughter F W 33 Married Housekeeper INKY VA Mirth PAINTER, GrandDaughter F W 5 Single MO KY IN Jessie PAINTER, GrandDaughter F W 2 Single MO KY IN
I couldn't locate matches in either the 1860 or 1900 census indexes that I have for either Samuel or Runnion.I did see some older queries on the RootsWeb Moore message boards concerning the Moore's in Cass County, Missouri.The email addresses may be in active, but you might give it a try.I also saw a few mentions of a Runyon/Runion/Runnion & Moore family connection from early Virginia.It's hard to say, but such an unusual first name might well be for another family's surname.If you want copies of the 1870/1880 images in pdf format, I could foreward them to you.Good luck with your search.