This year, for the first time, my husband also has some confirmation of his family lore, that he is descended from religious pilgrims who came to Massachusetts on the Mayflower. Through my genealogical research, I've determined he is, in fact, descended from at least six. Given how interconnected families became in the early days of Massachusetts, it's possible he's descended from even more.
Here is what I've managed to suss out to date.

John Alden and Priscilla Mullins
Not only is The Gryphon descended from John Alden and Priscilla Mullins, he is descended from arguably their most famous child: Elizabeth Alden. She was the first child of European descent to be born in Massachusetts, and reports are that she was born on the Mayflower. This makes a lot of sense, because the ship was actually moored in harbor for a number of months while those who were able -- and not sickly from the rough voyage -- found a place to settle and built a common house where everyone could live.
The connection to this couple works out as follows:
John Alden (1598-1687) m. Priscilla Mullins (1602-1688)
Elizabeth Alden (1624-1717) m. William Pabodie (1620-1707)
Martha Pabodie (1650-1712) m. Lt. William Fobes (1649-1712)
Elizabeth Fobes (1683-1737) m. William Briggs (1671-1751)
Judith Briggs (1710-1765) m. Jeremiah Wilcox (1683-1768)
Benjamin Wilcox (1747-1816) m. Patience Tucker (1746-1816)
Benjamin Wilcox (1785-1857) m. Patty Brownell (1794-1855)
Thomas Brownell Wilcox (1821-1908) m. Jerusha Ryder Smith (1827-1904)
Sarah Wilcox (1850-1940) m. William Henry Waterman (1845-1925)
Henry W. Waterman (1872-1932) m. Helen Clare Blake (1883-1963)
Henry and Helen were the parents of The Gryphon's maternal grandmother.

Francis Cooke and his son John
Francis Cooke was a signer of the Mayflower Compact, the governing document for Plymouth Colony. His wife, Hester, and their young children remained in Leiden, Holland, at a colony founded by English separatists, until the Plymouth Colony was established.
The connection to Francis and John is the following:
Francis Cooke (1583-1663) m. Hester Mahieu (1585-1666)
John Cooke (1608-1695) m. Sarah Warren (1614-1686)
Elizabeth Cooke (1645-1715) m. Daniel Wilcox (1637-1702)
Samuel Wilcox (1659-1702) m. Mary Wood (1664-1721)
Jeremiah Wilcox (1683-1768) m. Judith Briggs (1710-1763)
This Jeremiah and Judith are the same couple named above, with Judith Briggs a descendent of John Alden and Priscilla Mullins. So, as it turns out, both husband and wife were Mayflower descendants. Not an unusual circumstance in that part of Massachusetts, near Cape Code, as I'm discovering.

Stephen Hopkins and his son, Giles
Stephen Hopkins made the journey with his second wife, whose maiden name was Elizabeth Fisher. His first wife, Mary, possibly the daughter of Robert and Joan (Machell) Kent of Hursley, county Hampshire, was the mother of Giles. Elizabeth gave birth to a son, Oceanus, during the voyage to Massachusetts.
This line connects as follows:
Stephen Hopkins (1581-1644) m. Mary Kent (?)
Giles Hopkins (1608-1877) m. Catherine Whelden (1620-1688)
Mary Hopkins (1640-1700) m. Samuel Smith (1641-1697)
John Smith (1673-1717) m. Bethiah Snow (1672-1734)
Samuel Smith (1689-1773) m. Mercy Baker (1692-1769)
William Smith (1722-1761) m. Anna O'Killey/O'Kelly (1720-1761)
Obed Edom Smith (1755-1840) m. Abigail Paine (1754-1842)
David Smith (1788-1873) m. Jerusha Ryder (1794-1867)
Jerusha Ryder Smith (1827-1904) m. Thomas Brownell Wilcox (1821-1908)
Thomas and Jerusha are the same couple listed above, under the John Alden and Priscilla Mullins line of descent.
I could write much more about all many of the individuals listed above. Many of them became instrumental in the founding and running of various towns in Massachusetts and Rhode Island. Many members of these lines were surveyors, or served as judges during colonial times. Also of note, Benjamin Wilcox was a Captain in the Massachusetts militia, known as the Minutemen. Of particular interest to me, Henry W. Waterman was a newspaper man, working for The Boston Globe.
So happy Thanksgiving from me and my husband, the Pilgrim!