(Revised and corrected repost)
There are basically two incompatible versions of the origin of the Washington family.
The older, given out by Washington Irving in his biography of George Washington (vol. 1, 1855), cites a William de Herteburn who acquired the manor and lands of Wessington (in what is now county Durham) in exchange for his Herteburn lands, sometime prior to 1183. The evidence for this is a citation in the "Boldon Buke" (a survey compiled under the auspices and for the use of the Bishop of Durham, then one Hugh de Puiset): "Willelmus de Herteburna habet Wassyngtonam (Irving: Wessyngton), excepta Ecclesia et terra ad Ecclesiam pertinente, in escambium pro villa de Hertburna, quam propter hoc quietam clamavit, et reddit 4L., et vadit in magna caza cum ij leporaris, et quando commune auxilium venerit debet dare unam marcam (Irving: 1 Militem) ad plus de auxilio." (There are apparently four surviving copies of the MS, all with slight differences.)
That is, "William of Hertburn holds Washington, except the Church and the lands pertaining to the Church, in exchange for the manor of Hertburn, which he previously claimed, and he pays 4 pounds, and (difficult clause, seems to mean "he shall go on a grand chase with two hunting-dogs"), and when the [collection for] common aid comes [due], he owes an extra mark to the fund."https://books.google.com/books?id=1-g3AAAAYAAJ&q=wessyngton#v=o... (Irving's version has it that he owes a man at arms for communal defense, which is a bit more logical.)
Some curious additional information by way of MedLands has it that he may have been the third/final husband of Margaret of Scotland, daughter of Henry of Huntington, who was previously married to Duke Conan IV of Brittany and Henry de Bohun, 1st Earl of Hereford. http://fmg.ac/Projects/MedLands/SCOTLAND.htm#Margaretdied1201
Who was he to rate such a marital prize? Apparently he was connected to Northumbrian nobility, including the family that became ancestors of the Nevilles of Raby, though there is considerable uncertainty as to the exact nature of the connection.
The other version, which *totally erases the existence* of William de Herteburn, is a farrago of Nordicist nonsense published in 1879 (heyday of the crackpots) and tracing George Washington's descent from Odin(!) through Ragnar Lodbrokk(!!) and other mythical/legendary figures. Unfortunately the effect is thoroughly spoiled early on by the misidentification of Jarl Thorfinn Sigurdarsson of Orkney with the far less renowned Thorfin (or Thorkel) of Yorkshire. (They were *not* the same person, it virtually goes without saying.)
Thorfin of Yorkshire is cited in Domesday as having numerous holdings in 1066, but none thereafter. Presumably they were confiscated and added to the "Honour of Richmond" handed over to Count Alan "Rufus" de Penthievre (William the Conqueror's double second cousin), who then parceled out the goodies, some to sufficiently compliant previous holders (Thorfin not included),and others to relatives and followers - he gave some special plums to his own half-brothers, Bardolf, Bodin and Ribald (Eudes de Penthievre's sons by one or more mistresses).
This reign of Frogs apparently grated on Mr. Welles' sensibilities so much that he dismissed their ancestry as "errors" and re-fathered Bardolf in particular on Thorfin (still misidentified as the Orkney Jarl).
Bardolf did, in fact, have a son named Akaris (or Acarius), who was largely responsible for founding the abbey of Jervaulx, in Yorkshire near Ripon, and Acarius had a son Herveius or Hervey who became the ancestor of the FitzHughs (some generations later, with a prominent Hugh).
That wasn't enough for Mr. Welles, who added a Bondo fitzAkaris "of Wessyngton", with three sons who "just coincidentally" wiped over the period when William de Herteburn should have been in possession. (It's a classic "three brothers" story, and we should know by now how much trust to put in *those*.) People who go *that* far have been known to fabricate evidence to support their fictions, so it's anybody's guess whether the "charter of Bondo" found in Mr. Welles' book is authentic or not. (Irving mentions a "Bondo de Wessyngton" and son William - in 1257.)
The line of the Washingtons of Durham came to an end with Elenor de Washington, who married Sir William Tempest late in the 14th century. Their daughter, Dionysia (Denise) Tempest, married Sir William Mallory of the Yorkshire branch of that family (and they became the ancestors of the "Southern Mallorys" who fanned out from Virginia across the New World). A cadet branch of Washingtons, however, had previously moved into Lancashire and from there down into the Midlands....
Irving version here: William Fitz Patric de Washington
Welles version here: Bondo de Akaris