The birth records for the children in this family are all listed as Cock, but when the family moved to Kentucky, they apparently changed their surname to Cox, which is the spelling that appears on almost all records after 1870.

Carroll Co. marriage bonds record marriage of Peter Cock & Lucinda Phillips, but Joe Gadus records Peter's wife as Lucinda Cox. Perhaps she was a widow Phillips when she married Peter?

Either way, she was almost certainly a cousin of some sort to Peter. I wish I could find a death record for her or something else that listed her parents.

1850 census of Floyd Co., Virginia, enumerated 29 Aug, lists on p. 436 as family #817:
Peter Cox, 23, farmer, b. Carol; Lucinda, 20, b. Floyd; Luke, 1, b. Floyd; Sarah E., 2/12, b. Floyd.

1860 census of Floyd Co., Virginia, Indian Valley district, enumerated 27 July, lists on p. 123 as family #878/825:
Peter Cock, 30; Lucinda, 27; Elizabeth, 8; Virginia, 7; Amos P., 4; McNiely, 3; Lafayette, 2. All born in Virginia.
Since Elizabeth is listed in the 1850 census (making her barely 10), and Amos’s birth was recorded on 24 Dec 1853 (making him 6 1/2), the age implied by the 1870 census below is probably more accurate for Virginia, that is, she was probably 8 or 9.

1870 census of Elliott Co., Ky, Mocassin precinct, lists as family #187:
Cox, Peter, 48, farmer, Va; Lucinda, 40, Va; Elizabeth, 21, Va; Virginia, 19, Va; Amos, 17, Ky; McNeely, 14, Ky; Lafayette, 13, Ky; Nancy, 9, Ky; Ruth, 4, Ky; John, 1, Ky. (Although all the children from Amos on down are recorded as born in Kentucky, official birth records exist in Floyd Co., Va for Amos, McNeely, Nancy and Ruth.) The preceding household is his brother Tobias's son Maston.

I cannot find this family in the 1880 census (not even the children.)