The original Minutemen were semi-organized militias throughout the state of Massachusetts set up to protect villages and towns from the British during the American Revolutionary War. The term was created because they could be out and ready for a fight at a minute's notice. Though inactive for a couple of centuries after that, they regrouped in the 1990s to form a brief college basketball dynasty in western Massachusetts.
 Woodrow Wilson's Four Minute Men
...were not particularly lazy or slow Revolutionary War militiamen, but a propaganda outfit set up under Woodrow Wilson to explain why the United States should enter World War I, despite widespread isolationist sentiment among the U.S. public and Wilson winning re-election in 1916 on the slogan "He kept us out of war". The name (surprise, surprise) stemmed from the fact that the pro-war sales pitch would take... four minutes (give or take). Whether due to the Four Minute Men or not, public sentiment was roused with some hilarious results.
 Robert DePugh's Minutemen
The 1960s era Minutemen (no relation to the American Revolutionary War troops at all, nor to the later punk rock band) was an extreme anti-Communist paramilitary group in the United States founded in 1961 by Missouri pharmaceutical company owner Robert DePugh. DePugh was a one-time member of the John Birch Society who quickly decided the Birchers were a bunch of letter-writing wimps and the time had come to form underground paramilitary bands to take up arms against the Imminent Communist Threat. DePugh published a 1961 booklet on guerrilla warfare for the group, as well as a newsletter called "On Target". According to one source, the Minutemen were structured after the California Rangers, a 1950s paramilitary group founded by William Potter Gale, who later co-founded the Posse Comitatus.
The original idea was to have a network of Americans ready to wage guerrilla warfare against the Soviets after the commies invaded, but the group very quickly attracted the usual bunch of Really Unstable Individuals who were convinced the real communist threat was right here in the U.S. government. They were convinced that Washington was riddled with closet commies under every rock, lurking under their beds, and poisoning their precious bodily fluids. Nineteen members of the New York branch of the Minutemen were arrested in October 1966 for plotting to bomb three summer camps (though the charges were later dropped due to faulty search warrants), and members of the Seattle branch were indicted in 1968 when Seattle police infiltrated the group and uncovered a plot to bomb a hydroelectric plant and a Seattle police station.
DePugh himself was indicted twice on federal firearms charges in 1966 and 1968; the second charge also included conspiracy to rob a bank. The second charge stuck, and he went on the lam for two years before being convicted and spending 1970-1973 in prison. The Minutemen limped on a few more years sans Robert DePugh under new leadership; his brief replacement as head of the Minutemen was, of all people, a folk musician associated with a Scientology splinter group called the Process Church of the Final Judgment. Upon release in 1973, DePugh's book Can You Survive was published, which was supposed to be about how to survive while on the lam for two years but in actuality was mostly a bunch of paranoid International Communist Conspiracy woo. DePugh briefly went on to join Liberty Lobby, later converted to Christian Identity, then denounced politics entirely and dropped out of sight.
In terms of tactics, organizational methods and paranoia about foreign infiltration of the US, the Minutemen were a possible inspiration for the later militia movement. (Replace "communists" and "Soviets" with "the UN" and "the NWO".) However, apart from the name and the fact that both groups are right-wing and militant, there is no connection to the later border vigilante group the Minuteman Project.
 Jim Gilchrist's Minutemen
Today, the term has been appropriated by a group of vigilantes who stake out the border to catch illegal immigrants. Many of them are the same sorts of people who would have supported or joined the militia movement in the 1990s.