I did not want to be the one say it boxer (I already bore everyone enough with my Band ravings ;-), but you got it. Their name came from the residents of Woodstock, who came to think of Bob Dylan’s band as their own local Band (Woodstock is a small village in The Catskill Mountains of Upstate New York State. Perhaps you’ve heard of it, though The Woodstock Arts & Music Festival actually did NOT happen there).
While recording their debut album Music From Big Pink, the Hawks (they continued using that name, theirs since 1960, while serving as Bob’s road band) were trying to come up with a new name, vetoing amongst others The Crackers. Legend has it they were surprised to see the name The Band on MFBP when it was pressed and printed, but I find that hard to believe.
At any rate, The Band sure does fit them, though it should really be THE Band, they’re so much better than everybody else. IMO, there is The Band, and then there is everybody else. But ya’ll already know I feel that way!