This is kind of a trick question. In the original comic, there isn't actually a group of people who call themselves the Watchmen. The superheroes all belonged to teams called the Minutemen, named for Paul Revere's militia during the American Revolutionary War, or the Crimebusters. The comic takes its title from the phrase "Who Watches The Watchmen?" (which appears at various points during the book, including as graffiti in the background of scenes), a translation of the Latin "Quis custodiet ipsos custodes?," from Plato.
Essentially, "Quis custodiet ipsos custodes?" is a distillation of one of the main themes of Watchmen; namely, who will protect us from our protectors. Plato's response is that the protectors must police themselves, and should be taught that they are better than us so that the protect us out of a sense of obligation and duty. Moore and Gibbons' story not only shows us the heroes policing themselves through Rorschach's investigations (and ultimate discovery), but also at least two heroes who believe that they are better than us (Dr. Manhattan and Ozymandias).
In the movie, "Watchmen" becomes the name of the superhero team, because Latin-based allusions don't always fare well with mainstream audiences. Insert your own joke about Hollywood patronizing its audience here.