Lucy was the sixth and last child of the third president Thomas Jefferson and his wife Martha Randolph Jefferson. Her mother passed away just 4 months after Lucy's birth, due to childbirth complications.
This Lucy Elizabeth shall not be confused with her older sister Lucy Elizabeth, who passed away in 1781. She was born on 08 May 1782 at "Monticello" in Charlottesville, Albemarle, County, Virginia. She was named after her older sister who passed away 1 year before she was born.
Lucy Elizabeth Jefferson died when she was two years old from whooping cough on 13 Oct 1782 at Eppington Plantation in Chesterfield County, Virginia, USA. 
It may be possible to confirm family relationships with Lucy by comparing test results with other carriers of her ancestors' mitochondrial DNA.
Mitochondrial DNA test-takers in the direct maternal line: