In my quest to find Francis' lineage, I kept seeing 2 sources and finally was able to view them at the Tucker, GA Family History Center. After many hours of pursuing the leads from them, I found the following:
#1 Baker's History of Northamptonshire* for an extensive lineage of the Breton family living in Teton. This lineage, “Breton of Teton,” does show a Francis Brayton, s/o John and Jane Bosville Breton, but no dates or other information about him are given. John was baptized 3 Oct 1567, buried 27 April 1619. So it is unlikely that this the same Francis who was 'progenitor' of the Brayton line in America. Another Francis is listed, died 10 July 1659. Wife Mary Mulson d. 16 Jan 1704-5. But their children are not the right ones. This source may prove useful in finding related families.
#2 Passenger and Immigration Lists Bibliography, 1538-1900: being a guide to published lists of arrivals in the United States and Canada; Filby, P. William; p.39.
This is often cited for proof of Francis' immigration date. However the data is not contained in this book. Page 39 does list the supposed sources for Francis' immigration: material written by Samuel G. Drake. These 4 sources are part of the New England Historical and Genealogical Register and can be found online at Familysearch.org. Francis Brayton is not listed in any of these.
Vol. 1:2 (April 1847), pp. 137 – 139
Vol. 14:4 (October 1860), pp. 297-359
Vol. 14 (1860)
Drake, Samuel, Founders of New England, 3rd Edition, NEHGR, 1865, 161pp.
*“Baker's history of Northamptonshire, in Astor's Library in New York City, page 220, has a very long genealogy of the Breton families running back four generations before the first date 1197, showing that the Bretons were in England as early as 1060. The genealogy seems to have been prepared from papers of the family of William Le Breton in 1197. There appears to have been another page but it was copied backwards and is illegible. A hand written note on this same page says:
"This is C. B. Brayton's Singer Building, New York City, Executive Offices."”
If someone has found something different, please let me know. I just thought you'd like to know so you don't waste time in tracking them down.