See his father's memorial for a copy of William's will where Lewis and Harrison are named. Will

as per the 1850 census Barren Cnty KY.-
HEAD: "Harrison Poynter" age 37 farmer
wife Eliza J- age 27
children:
Margaret A- age 13 (B: 1837)
Sarah E- age 11 (B: 1839)
These 2 daughters were born to Harrison's first wife, nancy Dowdy-Poynter. She must have died in childbirth or shortly after.. Harrison remarried 1841 and had 3 more children by Eliza Jane:
Mary R.- age 7 (B: 1843)
William H. H. (William Henry Harrison) age 5 (B: 1845
Amanda J. age 3 (B: 1847)

sources: 1850 census-
"Kentucky Marriages, 1785-1920"
Harrison Pointer marriage record to Eliza Jane Payne
Event date: 21 Sep. 1841
Event place: Barren cnty. Kentucky
also:
Cuz' Melvin Schwartz (Poynter) many many years ago.
not sure where he gets the full name of William Henry Harrison Poynter SR the census records all reflect Harrison.

another Cuz' Susan Wilcox-Hosbach, who is also on my GREER family tree...NO ONE seems to have any proof of Harrison's first name being William or the middle name Henry? that is just what he and Elizabeth named their son William Henry Harrison not JR., I don't think.

And then from Sue Hosbach the "Administration of estate of HARRISON POINTER, Barren Co., Court book Vol. 8 pg. 312 LDS film #0,209-737-4 ....." So this is yet another legal document that reflects his name only as Harrison.

CHILDREN NOT LINKED (not found on f-a-g)
Margaret A. B: 1837 mother, Nancy Dowdy
I did find a marriage record: "Kentucky Marriages 1785-1920
Name: Elisha Arnold
Birth date: 1827
age: 26
Spouses: Margaret Poynter
birth date: 1837
age: 16
Event place: Barren,. KY
Event date: 18 Nov. 1853"

I can not find anything else on either of them.
no memorial for either made at this time.

the 2nd CHILD not LINKED is:
Sarah E. B: 1839 mother is also Nancy Dowdy.

found nothing on her anywhere...except the 1850 census living with her father Harrison.