Other spellings in primary sources: Doman, Dammant, Damond, Dammin
Birth: Before 1600.
This is a rough estimate based on his marriage in 1621. His parents and baptism has not yet been identified conclusively. The Damon Family of Reading, Massachusetts may identify him as the son of another Henry Damon - this source needs to be investigated. 
Notes on Life
Henry Damon died leaving two young children, likely as a young man himself. Parish records at Faversham, St. James in Dover, and St. Mary's in Dover have been used to conclusively prove his origins and that of his children.  His marriage and birth of his first child occured at Faversham; however, he soon thereafter removed to Dover where the baptism of his second child is found. After his widow Abigail remarried to John Eaton, his children emigrated with his wife on the Elizabeth and Ann in May of 1685.
Marriage and Children
He married Abigail Bachelor by license dated 6 March 1619/20 to be married at Faversham. 
'Henry Dammin, of Faversham, and Abigail Bacheler, s.p. (of same parish). At same. March 6, 1619'
Beware of a common error which names her as Abigail Gilson.
Jane Damon bp. on 15 August 1624 at St. James, Dover. She married John Plimpton on 13 March 1644 at Dedham. They had 5 children born at Dedham, and 8 born at Medfield, Massachusetts. John Plimpton was captured by Indians and burned at the stake in late 1677. She married second Nicholas Ide on 16 March 1680. They moved to Rehoboth where Nicholas was buried on 18 October 1690. She married third George Kendrick on 1 April 1691 at Rehoboth, Bristol, Massachusetts. Jane (Damon)(Plimpton)(Ide) Kendrick was buried on 12 May 1694 at Rehoboth.
Children of John Eaton and Abigail Bachelor
Mary Eaton bp. 20 Mar. 1630/1 at St. James in Dover. She married John Mason on 5 May 1651 at Dedham. They had 7 children born at Dedham. Mary (Eaton) Mason died at Dedham on 6 May 1676; John Mason died at Dedham on 18 January 1688.
John Eaton bp. 7 October 1632 at St. James in Dover; bur. 27 January 1635 at St. Mary's in Dover.
Thomas Eaton b. abt. 1634 (age 1 in 1635) in England; d. 10 September 1649 at Dedham.
Jacob Eaton b. 8 June 1642 in Dedham; d. 20 January 1645/6 in Dedham.
Death: Between August 1624 and 5 April 1630
Last known living 15 August 1624 when his daughter Jane was baptized.
Dead by 5 April 1630 when his widow remarried to John Eaton of Dover.
Footnotes and citations:
↑ Damon, Richard A. The Damon Family of Reading, Massachusetts, (Amherst, Mass.: R.A. Damon, 1999).
↑ Reed, Paul C. F.S.A.G. The English Ancestry of John1 Eaton, of Dover, Kent, and Dedham, Massachusetts. Research report commissioned by Joe Cochoit and the Eaton Family Association (2003). Available on the Eaton Family Association website (paid).
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It may be possible to confirm family relationships with Henry by comparing test results with other carriers of his Y-chromosome or his mother's mitochondrial DNA.
Y-chromosome DNA test-takers in his direct paternal line on WikiTree: