Thanks, Owen. I and those who have discussed this on soc.genealogy.medieval, have studied what is said in Robert Freville’s Will. in it, Robert Peyton is described as one of Rose’s friends or kinsmen, and no more than that. There is nothing to state what Robert Peyton's relationship to Rose was.
But, IF Rose was Anthony Hasleden's daughter, as is one of the possibilities and as is tentatively canvassed as possible in the entry for her son George in the History of Parliament, the evidence suggests that this would make Robert Peyton her kinsman: Anthony Hasleden had a niece Peyton, either a niece by blood or a niece by marriage - she is named as "niece Payton" in his Will, with no first name. See the soc.genealogy.medieval discussion.
In other words, Robert Peyton could be a kinsman of Rose whether her father was Thomas Peyton or Anthony Hasleden.
Visitation pedigrees always need to be treated with caution, and there are warnings in the WikiTree guidance on reliable sources. Contradictions can be found between Visitations for different counties - and, on occasion, between pedigrees for different families in the same Visitation. Many mistakes have been found in Visitation pedigrees. There are some salutary notes about Visitations on Chris Phillips’ very helpful site on medieval genealogy, here.
But in any event I have not found anything in Clay's edition of the Cambridgeshire Visitations which states that Rose was a daughter of Thomas Peyton. I may have missed something, but what was said on Rose’s profile reflected a misreading of a Burgoyne pedigree, which I have corrected today, adding a comment about this to Rose's profile. There is also an earlier edition of the 1619 Cambridge Visitation, edited by, and privately printed for, Sir Thomas Phillipps in 1840 - see p. 6, viewable on Google Books - this too is a Burgoyne pedigree which gives no information about Rose. What both editions say is that a Christopher Burgoyne married Thomasine daughter of "Robert Frevile of Shelford”, and that is all: they make no reference to Thomasine’s mother. Both editions have an earlier John Burgoyne marrying an unnamed daughter of a Thomas Payton: that is not relevant to this discussion.
It is worth mentioning that the original 2004 entry for George Freville, Rose’s son, in the Oxford Dictionary of National Biography, repeated a statement in the original late 19th century Dictionary of National Biography that George's mother was Rose Peyton. But this was revised online in 2008 to remove the statement that Rose was a Peyton. Instead the ODNB entry now gives her no family name. The author clearly decided that there is no firm evidence about who her father was. It is not just people like me, and those who have discussed the issues on soc.genealogy.medieval, who believe Rose's family origins are uncertain.
Please read carefully what is said in the soc.medieval.genealogy discussion, and the documentation that is set out there. I am not ruling out the possibility that Rose was daughter of Thomas Peyton. But there is no clear evidence to demonstrate this. It is just a possibility.
[corrected for typos; short paragraph added for clarification]
