I'm not related to this name at all, but heard today on a radio genealogy program that GAMESTER means "a female game keeper" (the ending -ster is the feminine, as in 'spin-ster'). So it is an 'occupational' surname.
There are only 43 of this surname in the USA phone listings, with 10 in NH, 8 in FL, 5 each in MI and CA, and 4 in IN. MA has 3, and AZ and ME have 2 each. CT, IL, OH, and OR have 1 each.
In Canada, out of a total of 25 surnames, fully 11 (44%) are in the smallest province - Prince Edward Island. Another 8 are in Ontario, 3 out west in Alberta, and 1 each in Quebec, Nova Scotia, and British Columbia.
All this suggests a connection between PEI or ON in Canada, and NH in the USA. Did this name start in NH and migrate outward, or perhaps did it originate in PEI and spread out from there ?