After Darin Mastroianni walked and Adeiny Hechavarria reached on an error in the first inning, Thames gave New Hampshire a 3-1 lead with his 14th homer.
The 2008 seventh-round Draft pick walked and grounded out before singling home Mastroianni in the sixth to snap a 7-7 deadlock.
The Fisher Cats broke the game open in the seventh, with Thames delivering a two-run single. He ended up scoring with Hechavarria on Shawn Bowman's 13th homer.
Thames' previous best was five RBIs on May 1, when he homered twice in a 10-9 triumph over Reading. The club record for RBIs in a game is eight, set by Brian Dopirak on May 8, 2009.
Over his last 12 games, Thames is hitting .440 (22-for-50) with 14 RBIs and 10 runs scored. That stretch has hiked the Eastern League All-Star's batting average to .286 with 61 RBIs, a total that ranks second in the circuit.
Mastroianni and Hechavarria scored four times apiece for New Hampshire.
Fisher Cats starter Rey Gonzalez surrendered six runs -- four earned -- on nine hits over five innings. Danny Farquhar (2-2) fanned three and yielded a run on one hit in three frames for the win.