The original Captain Avery was raised on a homestead with her mother and brother, Danny Avery, somewhere on the Island. She witnessed 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, with Avery becoming the unofficial leader of the motley community. 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 further 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.
When things got bad at the Avery homestead her brother Danny - now new owner of the homestead - wrote a letter to her requesting that she return home and help the remaining family and run the homestead as their mother had died and the Fog drew closer to the homestead daily. However, by the time the letter was sent, Olympia Avery was already dead and replaced by DiMA's synth; the plea for help going completely unanswered.
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.