The given name "Adaline" is used in the family, so "Ada" may be the correct name.

1.  LDS.  Family Search: Internet Genealogy Service:  IGI - International Genealogical Index (online at FamilySearch.org).

John GARDNER Spouse: Nancy Ann CORBIN Marriage: 17 Jan 1858, Clermont, Ohio Source: Marriage Records, 1801-1910. Ohio. Probate Court (Clermont County)

2.  1860 Census Images (online at Ancestry.com; Image #21 of 50):  New Richmond P.O., Monroe Twp., Clermont Co., OH, p. 432A/861, PN 21, 164/159, enumerated 11 Jun 1860, official enumeration date 1 Jun 1860 (extracted by Diana Gale Matthiesen).  For complete record, see Harbison McNEILL's family group sheet. 

John Gardner	26	M		Farmer			Ohio
Nancy   "	32	F		Domestic duties			do
Wm F    "	11/12	M					do
H McNeiles?	34	M		Farmer	8000	1200	do
[Harbison's wife, children, et al.]

Listed four households from cousin, James CORBIN. 

3.  1870 Census Index/Images (online at Genealogy.com; Image #44 of 54):  Batavia P.O., Monroe Twp., Clermont Co., OH, Roll 1181, Book 1, p. 197B, 337/326, enumerated 2 Aug 1870, official enumeration date 1 Jun 1870 (extracted by Diana Gale Matthiesen):

Corben Nancy A	83	F	W	Keeping House	8000		Maryland
Gardner John	30	M	W	Farmer		530	Ohio
_______ Nancy	36	F	W	Keeping House			Ohio
_______ Ada J	5	F	W				Ohio
_______ Nicholas C?	2	M	W				Ohio
Corbin Jane	48	F	W				Ohio

Nancy A. CORBIN is the mother of Nancy GARDNER and Jane CORBIN; they are living next door to Nancy A.'s son, Shadrach CORBIN. 

4.  LDS.  Family Search: Census Records: 1880 United States (online at FamilySearch.org):

Census Place: Ohio, Clermont, Ohio
Source: FHL Film #1255000; NARA Film #T9-1000; Pg 191C
	Relation	Sex	Marr	Race	Age	Birthplace	Occupation	Fa	Mo
John GARDNER	Self	M	M	W	42	OH	Farmer	OH	OH
Nancy GARDNER	Wife	F	M	W	52	OH	Keeping House	OH	OH
Ida GARDNER	Dau	F	S	W	14	OH		OH	OH
Nicholas GARDNER	Son	M	S	W	12	OH		OH	OH
Ruda GARDNER	Dau	F	S	W	8	OH		OH	OH

I checked the image online at Ancestry.com, and it is "Ruda."