I am recognized by the Mayflower Society as a descendant of William White (and others) through his son Resolved and although I have not done much research on Peregrine White and his desendants, I can tell you what the MF William White Family book Vol 13, which covers the first 5 generations, has to say: In the index there is no name of Byers. The are many "Mary Whites", most with husbands who are not Byers and many w/o husbands, many of which are unexplained. Page 16 lists Jonathan White (Peregrine 2, William 1) and his wife Hester Nickerson, daughter of Nicholas and Mary _____ Nickerson. This page lists 7 children for them, the last being Mary White b. prob. bef 1703 and on page 54 it shows her husband to be James Russell.
If there is anything else I can look up in the William White book, let me know. By the way, we have recently updated the Peregrine White Wikipedia page: http://en.wikipedia.org/wiki/Peregrine_Whitehttp://en.wikipedia.org/wiki/Peregrine_White