I found this:
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1. William^ Cotton of Strawberry Bank (now Portsmouth, N. H.),
on the "last day of March in ye year of our Lord God one thousand six
hundred & fifty," bought of Anthony Brackeit his dwelling house and
farm, " situate at Strawberry Bank,' next unto ye house of Walter Abbott
along by ye water side." This is the earliest record found of the ancestor
of more than half the New England Cottons. Of his origin we know
nothing. The deliberate destruction, by the early town officers, of the first
book of town records, in 1652, may have deprived us of information upon
this point.
Among the associates of Gorges and Mason in the Laconia Company,
under whose auspices were made the first permanent settlements at Straw-
berry Bank and vicinity, were John and William Cotton, merchants, of
London, sons of Sir Allan Cotton, Lord Mayor of London in 1625-6, and
their brother-in-law and nephew, Thomas and Eliezer Eyre. Though
neither of them came to New England, and their interest in the Company
was sold to Mason in 1634 (at which time William Cotton was deceased),
it seems surely a plausible supposition that William Cotton of Strawberry
Bank may have been of the same family.
Another theory connects him with the family of Sampson Cotton of
London, whose daughter Elizabeth was the wife of Edmund Sheafe, an-
cestor of the Portsmouth Sheafes. (See Waters's Gleanings, Vol. 1, p.
345.) No proof of either connection, however, has yet been discovered
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[https://archive.org/stream/cottonfamilyofpo00cott/cottonfamilyofpo0... Cotton family of Portsmouth, New Hampshire] (1905) by Frank Ethridge Cotton
This suggests there is no documentation, but they seem pretty clear that whoever he is, he isn't the son of the Lord-Mayor.
