Lucretia's parents were never married. She and her 4 siblings were all born out of wedlock and used their mother's surname. They couldn't use their father's name as he was married to another woman.
Sources
American Genealogical-Biographical Index (AGBI)
Name Lucretia Gibson
Volume 62
Page number 219
Reference Genealogy of Alexander Alvord of Windsor, Ct. By Sam. Morgan Alvord. Webster, NY, 1908. (823p.):345
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DNA Connections
It may be possible to confirm family relationships with Lucretia 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 Lucretia: