That being said if you think I'm wrong please let me know and provide your sources for any corrections. There's nothing worse that someone telling you you have it all wrong but not telling you why they think they are right. That being said, here is what I think I know about the de Say ancestry of Thomas Dudley.
|quartered gold and gules|
The farthest back this line can be somewhat reliably traced is to Jordan de Say. Jordan was married to Lucy de Remilly. William's parents were not the oft reported Geoffrey and Hawise de Clare. A Hawise de Clare did marry a Geoffrey de Say, but it was several generations later. The Jordan/Lucy lineage has been accepted by Keats-Rohan. It's not a nice tidy fit, if fact it make some assumptions with no facts to back it up, but for right now it's the best I'v got.
setting the scene
William de Say was born early in the 12th century. Henry I, of England, was crowned in 1100 after the death of his brother William Rufus. The following year, Robert Curthose, Duke of Normandy, attempted to invade England and claim the throne,he failed. In 1106, Henry invaded Normandy and imprisoned the Duke, his own brother, and took his title Duke of Normandy. This reestablished the Anglo-Norman kingdom. Many of the wealthy Norman lords held lands in both Normandy and in England.
Henry, unfortunately, had for his heir a daughter, Matilda. Before his death in 1135, Henry made the leading barons swear allegiance and support for his daughter. Matilda had left England at age five to marry Henry V, Holy Roman Emperor. Hence her title of Empress. Henry died in 1125 and Matilda married Geoffrey Count of Anjou. Despite their oaths many barons defected to the side of Stephen of Blois, a grandson of William the Conqueror. What followed was many years of war for England, in a period of time known as "The Anarchy." One of the barons who played a major role in this drama was Geoffrey de Mandeville.
In 1140 King Stephen was barely hanging onto his crown. He created Geoffrey de Mandeville the Earl of Essex. Kings and queens needed the support of their barons and to bind them to their sides they offered rewards of lands and titles. In 1141 Matilda was waging war in England and entered London. Geoffrey was in control of the Tower of London. He promptly switched sides and turned control of the tower over to Matilda who reconfirmed his title as Earl of Essex. Unfortunately for her, the people of London were against her and drove her out of the city. She fled to the city of Oxford and eventually to Devizes Castle until she finally gave up and returned to Normandy. Her son would eventually be crowned King Henry II of England. 
Concedo etiam quod Willelmus de Sai habeat omnes terras et tenementa quae fuerunt patris sui, et ipse et haeredes sui. Matilda was only in the position to write charters for a brief time. Why would she give William de Say his father's land? Jordan de Say was still alive in 1142. The chroniclers say that William was a stout and war like man , and Matilda was in need of fighters.  But he was also something else, he was the husband of Beatrix de Mandeville, sister to the Earl of Essex. Did the connection to the Earl influence Matilda's grants or did she even have a choice in the matter.
Beatrix and Geoffrey were the children of William de Mandeville and Marguerite, the daughter of Eudes de Rie. William's father, Geoffrey, had either come with William the Conqueror or came shortly there after. He was recorded in charters as early as 1070. Beatrix had previously been married to a Norman lord named Hugh Talbot. For unknown reasons, Hugh and Beatrix were divorced. It was written that Geoffrey brought his sister over from Normandy to marry de Say.  It is not known if Beatrix and Hugh had any children.
In 1143 King Stephen was back in control of London, and apparently he wasn't to happy with Geoffrey de Mandeville. Geoffrey was arrested in St. Albans in October of 1143 and stripped of his Earldom and his castles at Pleshey and Walden and the Tower of London. This loss of title and lands pushed Geoffrey into open rebellion. He set himself up as a 'robber baron' in the fenlands of England. He was joined by his brother in law William de Say. They seem to have set themselves up at Ramsey Abbey from where they could control and ravage the countryside. Geoffrey's actions were so terrible that he was excommunicated by the Pope. While besieging Burwell in Cambridgeshire, he received a mortal wound and later died at Mildenhall in Suffolk.  Although it was written that William de Say also died during this battle, he seems to have survived for a few more years. Presumably he sought forgiveness from King Stephen.
William and Beatrix had two children who lived to adulthood, both sons. William the elder and of course another Geoffrey. It is estimated that their parents were married by about 1135 and their father was dead by 1155. This would make them young men when they lost their father. But, men grew up fast and died young in those days, so it was not unexpected.
william of kimbolton
William's oldest son and heir held land in Kimbolton in Huntingdonshire and in Saham in Norfolk. William's uncle, Geoffrey de Mandeville has started a Benedictine Abbey called Walden near Saffron Walden. I'm sure it was to atone for past and future sins. The de Say family continued to support the Abbey with grants of land, and William's name is found in their records.  William's name is also recorded in the Liber Rubeus de Scaccario or Red Book of the Exchequer. This 'book' was begun by the officers of the exchequer in the 13th century and contains various documents, deeds, charters and other related matters. If William did anything of great importance in his life, it went unrecorded. There is not much more to add about him.
England had a new, stable king beginning in 1154. Henry II, the son of Empress Matilda was crowned. He was an energetic man who took his role very seriously. England's house was in order, at least for a while.
william's wife and children
The name of William's wife is unknown. She was not Aufrica, daughter of King William of Scotland. Aufrica married William son of Geoffrey de Say, nephew to this William. William had only two daughters as his heirs. The William who married Aufrica had a son, also named William, who was his heir.  If Our William was married to Aufrica then his daughters would not have been his heirs. Any hoo, William died at a fairly young age, by 1177, so he was about forty, give or take a few.
|all that's left of Ramsey Abbey|
Beatrice de Say was the elder daughter. She married Geoffrey FitzPiers who would be created the Earl of Essex by King John in 1199. Geoffrey's parents were Peter de Ludgershall and his wife Matilda. Matilda married Hugh de Bocland as her second husband.
Matilda de Say married William de Bocland, son of Matilda and Hugh. So it seems the two sisters were married to half brothers. See this blog article on the de Boclands. The lands of their father were first divided by charter and confirmed by King Henry II. This charter and division were reconfirmed by King Richard I in 1198. Beatrice got the lions share of her father's estate. This had more to do with her husband being a mover and shaker at court rather than the fact she was the older sister. Geoffrey was justiciar of England from 1198 to 1214 and pretty much got what he wanted.  Matilda got one manor, that of Bruninton, the service of two tenants and the promise of land to value of about ten pounds.
back to beatrix
Beatrix, mother of the two girls, lived a long time. She is said to have lived into her eighties. She was described as a "model woman" and was a frequent visitor to Walden Abbey, which had been established by her brother, the Earl. She lived at Rickling, Essex, five miles from the abbey. She feed the poor and sick from her home.  When Beatrix's nephew, the 3rd Earl of Essex, died childless in 1189 she was next in line. But in 1189 she was an old woman, so she passed tried to pass the title to her son Geoffrey. He could not pay the large amount of money he had promised to obtain the land and titles. The lands went instead to Geoffrey FitzPiers, husband of her granddaughter Beatrice, and a favorite of King Henry II. Geoffrey was given title by King John I when he inherited the throne in 1199.
 William Farrer and Charles Travis, Early Yorkshire Charters, Vol. 7, The Honor of Skipton, (Cambridge: Cambridge University Press, 2013)30-35.
 Timothy Venning, Normans and Early Plantagenets, (Barnsey: Pen and Sword, 2014).
 J. H. Round, Geoffrey de Mandeville, a Study of the Anarchy, (London, 1892), p. 233, quoting Harleian Cart., 54, I, 44.
 John Bernard Burke, The Roll of Battle Abbey, Annoted, (London: Edward Churton, 1848) 75.
 Dugdale Monasticon IV, Walden Abbey, Essex, II, p. 142. (When I plugged the line into my Latin to English translator it said that William de Say had an undaunted spirit and was warlike).
 Catherine Lucy Wilhelmina Powlett, The Battle Abbey Roll: With Some Account of the Norman Lineages, Volume 3, (London : J. Murray, 1889) 127.
 John Horace Round, Geoffrey de Mandeville, A Study of the Anarchy, (London: Longmans, Green and Co., 1892).
 William Dugdale, Monasticon Anglicanum: A History of the Abbies and Other Monasteries, Hospitals, Frieries, and Cathedral and Collegiate Churches, with Their Dependencies, in England and Wales; Also of All Such Scotch, Irish, and French Monasteries as Were in Any Manner Connected with Religious Houses in England, (London, 1823).
 Hurbert Hall, Red Book of The Exchequer, Volume 1, (London: Her Majesties Stationary Office, 1896). 373-374.
 Medieval Lands
 Geoffrey White, The Complete Peerage of England, Scotland, Ireland, Great Britian and the United States, Vol. 11, (London : St. Catherine's Press, 1949) 465.
 John Horace Round, Ancient Charters, Royal and Private, (London: Wyman and Sons, 1888) 108.
 Andrew Brown, Church and Society in England, 1000-1500, (Houndsmill, Basingstoke : Palgrave MacMillan, 2003).
What to know more about life in medieval England? This is a neat book with lots of good information.