Despite what the information below states, William was not an Earl of Leicester. In fact his death would have had to have preceded his elder brother Robert's death in 1204, because Robert was the 4th Earl of Leicester; and when Robert died, his holdings and title were held by his sisters and eventually the title ended up with Simon de Montfort through Amice/Amicia, Robert's sister. If William were still alive at the time of Robert's death, he would have become Earl of Leicester, and any male offspring would have become earls as well. Thus a male dominated descent from Robert de Beaumont is unlikely.
I look for William to be a descendant of one of Robert's sisters, perhaps Margaret (married Saher de Quincy) who had a daughter name Loretta who married William de Valognes of Scotland. Loretta and William perhaps had a son William, who may, even though born in Scotland, have inherited an estate near the now extinct village of Hamilton, Leicestershire and had a strong Scottish connection. He certainly could have visited Scotland around 1215, married a Scottish woman, etc. Certainly the William, father of Gilbert Lord of Buthernock, would have been born a lot later than 1156.
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The following I obtained from Karen Hamilton, World Connect db=alaskalady, rootsweb.com
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Information from Robert & Kathlynne Groner; 733 Carol Lane; Elyria, OH 44035; groner@@mediaone.com
Possibly born Hambledon, Buckinghamshire, England.
He was 3rd Earl of Leicester and is said to have been the founder of the illustrious house of Hamilton and to have assumed his surname in 1172 from the place of his birth, the Manor of Hambleden (Hamilton) in the parish of Barkby, County of Leicester. It is stated that about 1215, he went to Scotland to visit his sister [Margaret], the Countess of Winchester and her husband Sayer de Quincey, Earl of Winchester. Conflicts between England and Scotland made it necessary for him to return to England to save his estates from confiscation. The conflicting statements of noted writers' reference to the descent of the family of Hamilton from the Earls of Leicester leave in doubt whether Sir William de Hamilton was the individual member of the Bellomont family who was the founder of the house of Hamilton, but the belief that the family is descended from the Earls of Leicester is supported by the concurring statements of many genealogists.