I started my journey by first hearing that my father's mother was already a member of the Mayflower Society. How would I go about finding out about this? I did not grow up among my biological family and don't really know any of them. My parents are deceased as well as all grandparents. I have a copy of my father's birth, where his mother is listed by the "pet name" (Gertrude) used by my grandfather, rather than her birth name, Anna. She is listed by this "pet name" in a census as well. All other records (that I can find), shows her birth name.
Contact the Mayflower Society - on their website they give the local representative for your state. They gave me very explicit guidance and suggestions as how to proceed and what exact documents I would need.
Perhaps they can help with the issue of the difference between Gertrude and Anna on documents.