The following information taken from Medieval English Genealogy website at:
http://www.medievalgenealogy.org.uk/families/arg/argoutl1.shtml
The family originated from Argentan in the département of Orne, in Normandy, according to Keats-Rohan. (It had previously been suggested, in the Complete Peerage (vol.1), that they came instead from Argenton in Poitou.)
The surname occurs in documents in a bewildering variety of spellings. In the Domesday Book, it appears in the Latin forms 'de Argentomago' (Farrer) and 'de Argentomo' (VCH Bedfordshire), and in early documents the spelling is frequently 'Argentom' or 'Argentem'. Eventually - in fact, probably after the family itself had become extinct - it evolved into 'Argentine', which Hunter describes as a form 'more euphonious and more pleasing in every respect'. In these pages, except when quoting specific documents, the compromise form 'Argentein' will be adopted.
on David Argentein:
In Tudor times, no pedigree was complete without a Norman ancestor who had fought at Hastings, and frequently one would be provided even if the family had risen only recently from the yeoman class. When an elaborate pedigree was drawn up, in the reign of Elizabeth, for the Alington family of Horseheath, a descent was concocted, extending over three centuries, to connect the family with a fictitious Sir Hildebrand de Alington, later described as 'under Marshall to William the Conqueror' (Banks), and his son Alan, who was 'thought to be the chief doer for the building of Westminster Hall' under William Rufus (Burke).
More plausibly, another descent was given for the Argenteins, whose heirs the Alingtons were, to one 'Dauid de Argentonio', who was later called 'a Norman, and a martiall knight, who under King William the Conquerour, served in the warres' (Weever, apparently quoting Camden). The same David is shown, together with the arms later borne by the Argentein family, in a forged document known as the 'Tabula Eliensis', which purports to be a list of knights quartered on the monks of Ely by William the Conqueror (Topographer).
Although the surname appears in several versions of the late compilation known as the 'Battle Abbey Roll', it is impossible to know whether David de Argentein was among the Norman knights at Hastings in 1066: there is good evidence for only a handful of their names, and most of those come from the nobility (Camp). But he certainly existed, for he appears 20 years later in the Domesday Book, as a tenant in chief in Cambridgeshire and Bedfordshire. His holdings were small: in Cambridgeshire, one manor, in Croxton (of which he had been temporarily deprived by Eustace de Lovetot, sheriff of Huntingdon), 1 virgate, 20 acres in Caldecote and 1 hide in Westwick, and in Bedfordshire, 1 hide in Riseley. There does not appear to be any later record of the Argentein family holding land in any of these places, although they were active elsewhere in both these counties.
At this time the name David - like many other Biblical names - was evidently very uncommon both in Normandy and England. Only one other land-holder with this name is mentioned in the entire Domesday Book: David 'Latimer', or the Interpreter. Keats-Rohan suggests that this David, who was a tenant of William de Braose in Dorset, was probably identical with the David 'de Argentomo' of Cambridgeshire and Bedfordshire. William came from Briouze-Saint-Gervais, in the arrondissement of Argentan, and feudal relationships in Normandy were often replicated in England after the Norman Conquest.
In any case, David de Argentein's holdings in England were modest, and given the paucity of records in this period, it is not surprising that we know nothing more of him.