Elizabeth has been something of a dead end. Clearly she existed and married Joseph Wesson, but no other information is available.
In the 1810 census, there is a record of Robert Nicholson living in Isle of Wight County, Virginia. This is quite near Brunswick, where Joseph Wesson lived. Perhaps this could have been Elizabeth's father.
Sources
"United States Census, 1850," database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:MC6M-7JN : 9 November 2014), Joysey Wesson in household of Joseph Wesson, Shelby county, part of, Shelby, Tennessee, United States; citing family 1044, NARA microfilm publication M432 (Washington, D.C.: National Archives and Records Administration, n.d.).
"Virginia Marriages, 1785-1940," database, FamilySearch (https://familysearch.org/ark:/61903/1:1:XRQN-J3T : 5 December 2014), Joseph Wesson and Elizabeth Nicholson, 05 Sep 1813; citing Brunswick, Virginia, reference ; FHL microfilm 30,659.
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DNA Connections
It may be possible to confirm family relationships with Elizabeth by comparing test results with other carriers of her mitochondrial DNA.
However, there are no known mtDNA test-takers in her direct maternal line.
It is likely that these autosomal DNA test-takers will share some percentage of DNA with Elizabeth: