Since there seems to be a connection to the Everingham Sawmill, it is likely that Henry is a grandson of Thomas Everingham (b.~1703) or Thomas Everingham (b.early 1700s.) The names Thomas & Henry are common in both of these families.

probable child;

1. William (b.~1801 NJ)
   possible child;
   
2. Clayton (b.~1811)

   children of Henry & Rebecca;
   

3. Henry (b.Oct 18,1815)
4. Ellison (b.Apr 15, 1817)
5. Martha (b.May 20, 1819)
6. Aaron Radford (b.1824)

1. Unknown

Henry Everingham, later changed to Evernham, was described in some family records as having been married 4 times with Rebecca being his fourth wife. He had a total of 16 children according to the records, but only the last 4 with Rebecca were recorded. Sources for this are from a genealogical report prepared for Nat Guistina. According to a Mintle family report 'Mintle Memoirs'by Florence Nelson Mintle, Henry Evernham was the owner of a sawmill at Toms River NJ. The original sawmill was destroyed by the British during the Revolutionary War, but Mrs. Mintle doesn't know if Henry inherited the sawmill or purchased it. The only other name linked to the sawmill is Thomas Everingham who had land surveyed near the mills location. In the same report it states that Henry came from England in 1776, according to Gertrude Evernham, a granddaughter of HenryIII (no proof). Henry lived close to Benjamin Everingham (1764-1850) in West Windsor NJ and is thought to be a relative of his. Some of his children & grandchildren migrated to Iowa along with grandchildren of Benjamin Everingham (Lydia Everingham Pierce)

Research of Kevin Everingham; (research information)
Book Note ... I believe HENRY is the "grandfather" Everingham mentioned in the book: "Fare to Midlands" by Henry Charlton Beck 1939. The book mentions; George, William & Ellison Everingham. ... The book says; "George Everingham, whos father and grandfather were smiths & wheelwrights before him at the crossroads at Red Tavern (Applegarth)..." George's Father was Ellison which would make this Henry the grandfather talked about in the book, and this means he worked as a "Smith & Wheelwright".