I don't have the ancestry of Anna, but I do have a copy of E. Charles HINCKLEY's Hinckley Heritage & History, 1982.Perhaps one of the clues below might help to make a connection.
We have a few differences in our HINCKLEY data.This is what I have:Henry E. ROOT m. Anna Mae HINCKLEY (19 Aug. 1856, Pittsfield, Lorain Co., OH - 16 Jan. 1929, Omaha, Douglas Co., NE; d/o Daniel W. HINCKLEY & Elizabeth _ [1828 - 1914, Oberlin, Lorain Co., OH]).
FamilySearch database of the 1880 Census, Avon, Lorain, OH:ROOT, Henry E., 23, station engineer, OH OH OH; wf. Ann M., 23, OH NY OH; ch. Carlton H., 2, & Florence M., 10 mos.[I don't have dates for the other children you listed:Hamilton E., Hattie A., Ada B., & Ida B.Were the last two twins?There seem to be too many births, four more children born before the second marriage in 1885.]
Pittsfield, Lorain, OH, 1880:HINKLEY, Artemon, 35, OH NY NY; wf. Emma, 38, OH MA MA;ch. Cora, 14; Willis, 8; Edward, 6; Libbie, 4; Gersham, 2; & Ida, 1. [There were earlier Gershoms of Pittstown, NY (near southern Vermont), & Rome, Oneida Co., NY.]
Grafton, Lorain Co., OH, 1880:HINCKLEY, Dexter, 49, MA MA Canada; wf. Rhoda, 51, OH. [In spite of the Massachusetts birthplaces, he might be the son of Coleman HINCKLEY, of Cape Vincent, NY, & Isabella BRADSHAW.]
Ridgeville, Lorain Co., OH, 1880:M. Edward HINCKLEY, 38, OH; "others."
Possibly the Lorain Co. family connects to the nearby Portage County family. From Hinckley Heritage & History:Benjamin HINCKLEY (23 N 1782 - 3 May 1835) m. 10 F 1806, Willington, CT, Susannah DAVIS (3 Jan. 1783 - 8 Jan. 1873, Hiram, Portage Co., OH).They had
Susan Harris HINCKLEY, b. 12 Apr. 1807, m. 5 Jan. 1827, Hiram, OH, Arial PROCTOR.Susan was 73 in 1880, Hiram, OH.
Ann H. (16 F 1810 - 28 July 1870, Farmington [Trumbull Co.?] OH) m. 14 May 1834, Moses BUNDY
Eber Davis (10 July 1812, CT _ ) m1. Nancy JOSLIN; [m2. 17 D 1863, Adeline/Angeline COX?]
Sophronia (13 Oct. 1816 - 6 Oct. 1881, Rochester, Lorain, OH) m. 31 Aug. 1837, Wm. H. MARSHALL (ca. 1820, PA - _, OH).Rochester, 1880:Wm. H., 69, OH PA PA; Sophrona, 64, OH CT _; ch. Mary, 26, & Eugene, 23; dau. Eliza TANNER, 34, OH, & her son Hiram H. TANNER, 10.
History of Hiram, OH, says Benjamin HINCKLEY (22 Nov. 1781, CT - 8 May 1835) m. Susannah DAVIS (2 Jan. 1782 - 8 Jan. 1878).They moved to Ohio in 1813, with [daughters & son] E. D., b. 10 July 1812, Tolland Co., CT, who m. Nancy JOSLIN.
Hiram, Portage, OH, 1880:Eber D. HINCKLEY, 67, CT CT CT; wf. Angeline, 55, NY VT CT; son Eber P., 29; Henry, 21.
Sorry the clues got so scattered!Do you see any connections?