It would seem that the most acceptable term is "Connecticuter," which is defined by several dictionaries to mean "a resident of Connecticut." Dictionary.com even goes so far as to include "Connecticuter" on its list of offical names for residents of each state.
According to the History and Genealogy Unit of the Connecticut State Library, however, "There is not any nickname that has been officially adopted by the State for its residents." In their document on Connecticut's Nicknames, they mention several other terms that have been used in print to describe someone from Connecticut, including "Connecticotian," by Cotton Mather in 1702 and "Connecticutensian" by Samuel Peters in 1781. Wow; that's a mouthful!
Of course, there are still some who insist on calling people from Connecticut "Nutmeggers." This nickname, while certainly easier to pronounced than other alternatives, seems awfully old-fashioned. While Connecticut has been called the Nutmeg State, its official nickname has been "The Constitution State" since 1959. Plus, there is no definitive explanation as to how Connecticuters got themselves intertwined with the aromatic spice.
Confused yet? There's one more term to toss into the mix--"Connecticutian." Oddly enough, when I moved to Connecticut in 1996, this was the first term I heard used for someone from Connecticut. "Connecticutian" even shows up in some dictionaries as a noun meaning "an inhabitant of Connecticut."
So, what should you really call someone from Connecticut? "Connecticuter" gets my vote, but others from Connecticut may feel differently. You can honestly use any of these terms without offending us. Just don't call us late for dinner. Connecticut is, after all, the Land of Steady Habits.