About half of the 102 original Mayflower passengers died within the first year. So everyone claiming Mayflower ancestry has to trace their lineage back to one of the survivors (some of those who died had children, but the pool of possible ancestors is limited to a list produced by more than a hundred years of research. The "official" keeper of the list is the General Society of Mayflower Descendants (GSMD). Membership in the Society would seem pretty simple; all you have to do is prove a direct line connection with one of the people on the list or any of their descendants. That may sound simple, but I had one person during the past week who claimed to be a descendant and it turned out the person her family claimed as an ancestor died without any provable children and was not on the list.
If you would like to get into the battle over whether or not one or the other of the people in the Plymouth Colony were descendants of the original boatload, then be my guest. It could turn out to be a full time job. Just analyzing the existing source material is an overwhelming task. Fortunately, all of the huge amount of research that has gone into establishing the historicity of these people is recorded in an encyclopedia-type set of books called the following:
Kellogg, Lucy Mary, Edna W. Townsend, and Robert S. Wakefield. Mayflower Families Through Five Generations: Descendants of the Pilgrims Who Landed at Plymouth, Mass., December 1620. Plymouth, MA: General Society of Mayflower Descendants, 1975.
There are about 30 or so volumes of this series, each volume dedicated to one or more of the passengers and their descendants. As I found out, research on these lines in ongoing and there was a major correction to the Richard Warren line since I last updated my database. So, I had some extensive revisions to my outdated data.
Originally, this project was supposed to trace the ancestry of Thankful Tefft, but because no parents have been located for either Francis Cooke or Richard Warren, I decided to correct my data coming forward while I had the time and the resources to do so. I had researched the connection previously to my satisfaction, but with age comes increased skepticism and I needed to redo the research. It turned out to be a good thing to do since I had not kept up on the latest discoveries in my lines from the Society and hence the extensive revisions.
Essentially, the current status of the research has discredited the identity of the wife of Richard Warren I had in my records and now there is a new person identified as the wife and accepted by all of the people who matter.
So using the materials in the Family History Library, I was able to document from Thankful Tefft to John Cooke, a Mayflower passenger, who married Sarah Warren. John Cooke was the son of Francis Cooke, also a Mayflower passenger and his wife was the daughter of Richard Warren, a passenger. Richard Warren's daughter came over to Plymouth on another ship after the Mayflower.
I plan to put all this current research into TheAncestorFiles.blogspot.com when I get some time.
Normally, I would have no interest at all in trying to connect with someone famous. I have no desire to belong to a descendants' organization even though I could belong to all sorts of them. But I hope you can see that getting into this area of genealogical research can be taxing in the extreme. Most of the research on the 5 generations of descendants is cast in stone, so to speak, and the only way you could advance an argument would be through finding some additional original documents.