The real Captain Avery was the informal leader of the Far Harbor community and its villagers. The rugged, capable Captain Avery had lived on the Island her whole life and has more than one story to tell about it. She saw the Fog grow thicker and thicker, until islanders were driven to the Mariner's dock on the shore, which eventually became Far Harbor as most know it. Avery had been helping out ever since, becoming a pillar of the community. However, she also spoke out against the Children of Atom and their presence.
To calm the people down and prevent conflict, DiMA killed Captain Avery and altered a synth to look exactly like her and take her place in Far Harbor, even to the point where the synth would believe she is the real Captain Avery by modifying her memories. As expected, the duplicate would be open to the idea of the Children of Atom being on the Island.
The following is based on unverified behind the scenes information and has not been confirmed by canon sources.
Her name is most likely inspired by the real life pirate captain Henry Avery, linked to the story of the likely fictitious pirate colony already referenced by Bethesda by way of the raider stronghold Libertalia.
End of information based on unverified behind the scenes information.