Your post caught my eye because I have a William H. Nichols who married a Catherine Foley (might her name have been "Mary Catherine"?) in Meigs County, OH on June 18, 1868. [Source: Marriages of Meigs County, 1819-1913 and Ohio Marriages, 1800-1958]
Catherine was born in 1849 and her parents were Thomas Foley (1808-1900) and Elizabeth Wilson (1814-1894). Thomas and Elizabeth were living in Vinton County by 1870 and previously were in Meigs County.
A son, Andrew Jackson Foley, of Thomas and Elizabeth Foley is living next door to a James (1804-1870) and Mary/Mazie (1814-1890) Nichols in 1880. I believe this to be William's family.
The William Nichols who married Catherine Foley was born Feb 1845 and died Apr 6, 1926. And, yes, William and Catherine Nichols did have a daughter Mary E. Nichols born in 1869.
1880 United States Federal Census Name: William Nicolas [William Nichols] Age: 35 Birth Year: abt 1845 Birthplace: Ohio Home in 1880: Salem, Meigs, Ohio Race: White Gender: Male Relation to Head of House: Self (Head) Marital Status: Married Spouse's Name: Cathrine Nicolas Father's Birthplace: Ohio Occupation: Farm Laborer Household Members: Name Age William Nicolas 35 Cathrine Nicolas 32 Mary E. Nicolas 11 Melvin Nicolas 9 Owen S. Nicolas 7 Rohda Ann Nicolas 4 George L. Nicolas 3 Emma J. Nicolas 1
1870 United States Federal Census William Nichols Age in 1870: 25 Birth Year: abt 1845 Birthplace: Ohio Home in 1870: Columbia, Meigs, Ohio Race: White Gender: Male Post Office: Lee Household Members: Name Age William Nichols 25 Catharine Nichols 22 Mary E Nichols 1
1860 United States Federal Census Name: William Nichols Age in 1860: 15 Birth Year: abt 1845 Birthplace: Ohio Home in 1860: Columbia, Meigs, Ohio Gender: Male Post Office: Pomeroy Household Members: Name Age James Nichols 55 Magy Nichols 46 William Nichols 15 Eliza Nichols 13 David Nichols 10 George Nichols 8
1850 United States Federal Census Name: James Nichols Age: 46 Birth Year: abt 1804 Birthplace: Virginia Home in 1850: Columbia, Meigs, Ohio Gender: Male Family Number: 1253 Household Members: Name Age James Nichols 46 Mary Nichols 36 Samantha Nichols 21 Phidela Nichols 21 Franklin Nichols 19 Anna Nichols 18 Rachael Nichols 16 Baden Nichols 16 James Nichols 14 William Nichols 6 Eliza Nichols 4 David Nichols 0
U.S. and International Marriage Records, 1560-1900 Name: Mazie Mcglaughlin Gender: Female Birth Place: OH Birth Year: 1814 Spouse Name: James Nichols Birth Place: OH Spouse Birth Year: 1804 Marriage Year: 1843 Marriage State: OH Number Pages: 1