Upon further scrutiny of the Lawrence County Census records, I am inclined to suspect that the T. F. WILLIAMS shown in the 1880 Census record is probably Thomas F. WILLIAMS, age 33 (b May 1867 - TN) enumerated with wife Minnie, age 23 (b Sep 1886 - TN) and daughter Virnie L., age 2 (b Oct 1897 - TN).That Thomas F. WILLIAMS is shown to be born in TN to a father born in AL and a mother born in TN.He is also shown to be a Blacksmith.This seems unlikely to me to be a coincidence.
Source:Year: 1900; Census Place:Civil District 4,Lawrence,Tennessee; Roll:1584; Page:20A; Enumeration District:0052; FHL microfilm:1241584.
A T. F. WILLIAMS is shown to have married Minnie TITSWORTH in Lawrence, TN, on 22 Sep 1891.
Source:Ancestry.com. Tennessee State Marriages, 1780-2002 [database on-line]. Provo, UT, USA: Ancestry.com Operations Inc, 2008.Original data: Tennessee State Marriages, 1780-2002. Nashville, TN, USA: Tennessee State Library and Archives. Microfilm.
A death certificate for a Tom WILLIAMS (b 09 May 1867 - Giles, TN, d 09 Aug 1935 - Giles, TN) identifies Toms father as Bob WILLIAMS and his mother as "Mirah ROPPER".
Source:Tennessee State Library and Archives;Nashville, Tennessee;Tennessee Death Records, 1908-1959; Roll #:7.
It is UNCLEAR whether the assertion on the death certificate that Tom WILLIAMS was born in Giles County is correct, but this suggests another place to look for Robert and Telitha Roper WILLIAMS in the 1870 Census.