Robert -
Thanks for all you're doing on the Stiles and for starting this g2g. Your one name project's sources were helpful as I looked into this question.
Let me start by answering the question and saying I have note been able to find a reliable source and would further conclude that the sources I can find suggest Thomas was not a son of Edmond Style by Mary Berney.
As WikiTree stands today, Thomas Stiles would have children (like Francis Stiles-26) who appear to be Gateway Ancestors, as they show Magna Carta and royal lines through Thomas' mother, Mary Berney-17. Given that Francis Stiles doesn't appear in any of our research, we would classify him as a "Questionable" Gateway (see [[Category:Questionable_Gateway_Ancestors]] for our sub-project, where we review Questionables).
As a bit of background, I started reviewing this Magna Carta and royal line with Margery Despenser-26 and then through her Wentworth issue. Mary Berney-17's grandmother was Margaret Wentworth-21. Margery Despenser-26 and Sir Roger Wentworth are well known, covered by Cokayne's Complete Peerage, Douglas Richardson's trio of Royal Ancestry series, etc. Margery Despenser-26 is a known ancestor of at least a dozen or so well researchrd Gateways and is somewhat popular as a result.
In Royal Ancestry, by Richardson, Vol. 3, p 406, he shows a profile of John Berney-16, Esq. (Mary Berney-17's father), who married (1st) Margaret Reade and (2nd) Alice Ferrer. In this he mentions Mary as the wife of William Dennis, Esq. (This is a change from Richardson's earlier work, where he had shown her as married to Robert Jenney. Richardson has changed the Mary who married Robert Jenney to her aunt.)
John Berney-16 left a will dated 22 Jul 1553. In this will, Mary is mentioned and would appear to still be unmarried. (http://familytreemaker.genealogy.com/users/b/o/l/Andrew-N-BOLS/WEBSITE-0001/UHP-0110.html)
I have also seen the marriage record of a Mary, daughter of John Berney, Esq. who m. Edmond Style born 27 Mar 1538 on Mary's profile. And sources like Stiles Family in America p 706 that show her having issue.
So, in relation to this line, my first question would be do you think you have the correct Mary?
Moving on to Thomas:
Well sourced reference for Edward does not show him having a son Thomas:
http://www.archive.org/stream/stilesfamilyinam00stil#page/704/mode/2up
Neither does Edward's pedigree:
http://theweald.org/d10.asp?bookid=WBK412
Then, as has been pointed out on Thomas' profile and what I think you guys were discussing early in this post regarding what Styles Family in America has to say about Thomas origins, basically "not much is known about Thomas... he was a carpenter in Milbrook, Bedfordshire."
Some other notes regarding dates: Given the father Edmond was born 1538 (per his marriage record), then Thomas' profile says he was born 1550(?). Mary lived to age 98? Born when her father was 18 in 1828? When did she and Edmond get married? Edmond lived to age 12, yet had mulitple children?
I don't think there is sufficient proof at this point to have Thomas connected to Edmund and Mary and I would still have questions about Edmund and Mary.