There was another Cockermouth.
In New Hampshire USA is a river the Cockermouth River.
Wikipedia gives some information as to the reason for our local town having a namesake.
It seems that the town of Groton was originally named Cockermouth in honour of Charles Wyndham, who they say was Baron Cockermouth and Earl of Egremont, and was Great Britain's Secretary of State for the Southern Department from 1761 to 1763.
The town may no longer be called Cockermouth, but the Cockermouth River still flows past.
Due to non-settlement the land was regranted in 1766, then renewed in 1772. In 1796, one of the later grantees, Samuel Blood, succeeded in renaming the town after his hometown, Groton, Massachusetts.