Uncle Archibald Thweatt

Archibald Archer Thweatt (1876-1956) is the father of Archie, Frances, Libby, Anne, and Margaret. He is located at the bottom of the Thweatt family tree shown below. His Great Great Grandfather John Thweatt (1745-1802) had a son named Archibald (1772-1844). (You'll seem them at the top of the tree below.) This Archibald was the Great Great Great Uncle of Anne and Margaret. He provides the connection to Thomas Jefferson via his marriage to Lucy Eppes.

Jefferson Connection

Great Great Great Uncle Archie (1772-1844) married Lucy Eppes whose brother was John Wayles Eppes (tree below). This marriage provides one connection to Thomas Jefferson, since John Wayles Eppes married Maria Jefferson, youngest daughter of Thomas and Martha Jefferson.

But there are also two direct connections to Thomas Jefferson that do not depend on the sibling relationship between Lucy Eppes and John Wayles Eppes. 

• Both Lucy Eppes and Maria Jefferson were granddaughters of John Wayles (via different wives of John Wayles). Lucy's mother was born of the marriage between John Wayles and Elizabeth Lomax. And Maria Jefferson's mother Martha Wayles Jefferson was born of the marriage between John Wayles and Martha Eppes. So Lucy and Maria Jefferson were first cousins in virtue of a common grandfather (John Wayles).
• Additionally, both Lucy Eppes and Maria Jefferson are the great granddaughter of the father of Richard and Martha Eppes. So Lucy and Maria are also 2nd cousins by virtue of shared great grand parents (in the Eppes line).

Summary: The marriage between Archibald Thweatt and Lucy Eppes ties the Thweatt line to Thomas Jefferson in three different ways.

Fitting Sally Hemmings into the picture

On January 28, 2018, the New York Times Review of Books carried a review of a new book by Mary Beth Norton with the title Jefferson’s Three Daughters — Two Free, One Enslaved. The following paragraph appears in the book review ...

The lives of four Virginia families — Jeffersons, Randolphs, Eppeses and Hemingses — were intertwined. Sally Hemings, daughter of Jefferson’s father-in-law by his mulatto slave Elizabeth Hemings, arrived in the Jefferson household in Paris in 1787 at the age of 14 assigned to be the child Maria’s companion. Following Hemings’s oral history, Kerrison relates how Sally, knowing she could claim freedom by remaining in France when Jefferson returned home in 1789, negotiated a promise of eventual freedom for herself and all her children, who were by parentage seven-eighths white — a promise kept in part by Martha Randolph after her father’s death in 1826. Jefferson listed Beverley, the oldest boy, born 1798, and his younger sister, Harriet, as “run” in 1822, but other sources reveal that he quietly facilitated their departure.

-- NY Times, Jan. 28, 2018

The Eppes family referenced in the above paragraph is the family of Lucy Eppes who married Archibald Thweatt. Lucy's brother John married Maria Jefferson. Sally Hemmings, the African-American daughter of Thomas Jefferson, is identified in the paragraph above as Maria's childhood companion in Paris in 1787 when Maria and Sally Hemmings were children. Martha (also mentioned in the paragraph above) is the older Jefferson daughter and the sister of Maria and half-sister of Sally.

A detailed look at the three known connections to Thomas Jefferson

This video explains the ties between the Thweatt family and the family of Thomas Jefferson.