William Beamon (variantly spelled Beman, Beamond) was an early settler of Saybrook, Conn. He came aboard the ship Elizabeth in 1635, with a John Beamon who is conjectured to be his brother. Born ca. 1608 (age on the London Port Book was 27) and hailing from Bridgenorth, Shropshire, since that is where the magistrates were from who swore his allegiance. That last bit was from research done, yet again, by Mary Walton Ferris, in her Dawes-Gates Ancestral Lines [2:116-221] Despite that clue, along with two men of the same name traveling together, the English origins of William Beamon is still not known.
Sadly, Ferris says an extensive search was taken for the name Beamon in Bridgenorth, but nothing was found. She then veers north and notes a "family story" that says the Beamons were from the north. She recounts that there was a known relationship between Beamon and Richard Saltonstall, although two of her three examples she admits are untrue. Saltonstall was a founder of Saybrook where Beamon eventually moved by 1643. However, to presuppose a relationship is a stretch. Beamon first goes to Salem as does the John Beamon, who remained there.
William married at Saybrook on 9 December 1643 (Saybrook VRs, p.3), Lydia Danforth, the daughter of Nicholas and Elizabeth (Barber) Danforth. They had seven (possibly eight) children, including Elizabeth Beamon, wife of John Chapman. [All births of the children but one are in the Saybrook VRs]
Additonal sources: Anderson, Sanborn & Sanborn, The Great Migration: Immigrants to New England 1634-35, I:220-22.
Some more thoughts: Gamaliel Beamon also came to New England in 1635 (on a different ship) and settled in Dorchester and then Lancaster, Mass. He is of no known relationship to John and William, but how interesting that three men of the same name traveled in the same year to New England. Does that point to Beamon being a more common name than we think? Also, be aware that there is a Bemis family in New London, Conn. next door to Saybrook, but they are a totally separate family. However, I've seen people mistake the two names when reading colonial records.