I did not think about a connection being anything but a co incidence until I saw you? over on the Mason board. If you are still looking for a connection between Captain John Mason and Foliot, Meacham, Gould and Very, I will share what I know and you can decide whether it is possible or no. If all of the name Folliot were decended from a single ancestor of that name in Devon then you would look at the history of the families of de Redvers, de Courtney and de Vere - Earls of Oxford and their relation to each other by marriage. (Its a Roger Foliot of Normandy that supposedly is the ancestor who started the Foliot of Devon?)
I am investigating the chance that Lion Gardiner knew both Jeremiah Meacham and Captain John Mason ( if the John Mason at Saybrook was the same man and not just another of the same name) Mason and Meacham could be another more likely relation as both surnames could mean basically the same thing in my opinion. Meacham, Micham is Old English - a big ham or estate or settlement or house of and that would be not a stone mason but a Maison or Masson as the french might say? There were de Michams in London in 1272 according to The London Eyre of 1272 www.british-history.ac.uk/report
According to one of the first surname books I read, the only source for the surname Meacham would be a place name and the village of Mitcham in Surrey.
The coat of arms that was the source of a heated debate on this forum bears a similar design to the one for Captain Mason. So somewhere in the past it must have occured to others that there was a relation between Mason and Meacham.
Anyway I suggest if you have not found an answer to your question try this if you have not:
PRO England web site and under search Access to Archives
Run all the surnames through there that have to do with the Folliots and see what comes up.
Captain Mason had no male heirs just a daughter who married a Tufton and had two grandsons who took the name Mason and apparentley settled in Mass.