Sorry it has taken some time to reply to you, but chaos reigns supreme here at the moment. Everingham is a small village in Yorkshire, it lies just off the edge of the Ermine Street north of the river Humber. Ermine Street has been used by all who wish for rapid transport from the north of Britain to the south.
Templars understood this well and hence many of the Templar lands adjoined Ermine Street, it was their main route to London: Everingham village was on Templar lands.
The Everingham family most probably entered Britain at the time of the "Conquest" 1066, "Battle of Hastings" and all that. The supporters of William were granted lands (land usurped form the Saxon warlords after the conquest). Everingham's were more than likely to have been just that, ie. supprters of William the conqueor. They would have been granted land which they could then call "their own" but held in tenure of the king.
Robert de Everingham held in trust much land in Lincolnshire, including the Manor of Rowston, which he gave to the Templars of Temple Bruer, he was most likely a Templar himself, either active or as a pensioner (kind of retainee of the Templars, --- who knows there is no record of this explicitly). It is interesting to note that Temple Bruer was instigated by one Lady Elizabeth de Calz (a Norman family), who also gave land from Rowston "On land given for
the purpose by William de Ashby" (Mss. B. M. 4936) about the year 1134.
A second charter has land donated by Lady Matilda de Calz to the "Templum de la Bruere" prior to the year 1275 when "the Priory at Haverholme held land in this vill" The year that this happened? Well, Temple Bruer was begun in the year 1134 AD according to some authorities, "White's Directory" probably one of the best and most accurate, but I have also seen doccuments which say that the Temple was not begun until 1154 or a little after (Notitia Monastica p.274). As for the death of Robert de Everingham, 1215 seems to be a good bet, again from "White's".
Under the record for Rouston (Rowston) a Robert de Everingham died in the year 1287 as "Lord Paramount" of Rouston, but obviously this was a later Everingham than the one spoken of above.
Obviously it is important to undertsand that there were no "surnames" or "Sir Names" in Britain at this time, the term "Everingham" being to distinguish between "Robert's" from different places, viz Robert de Ashbei and Robert de Everingham Robert Textor (the weaver), Robert Couper/Cooper/Cowper (the barrel maker), etc.
I really hope that this helps, there are other records in Lincoln; Bishop Suttons records, etc. that may turn up something and I have to read through these a little later. If they turn up anything I will get back to you.