This is not copyrightable. All that you can do for an artist/band name is servicemark it, which is basically way that people won't confuse two different people offering services. The best way to go about doing this is just to get your name out there - start playing shows. If there is a local band in New York and a local band in Boston, both with the name "The Repeaters," there is not a big chance of the two confusing the two, and the legal system won't do anything about it. However, if you're going to name your band Aerosmith and you're from Boston, there is a big chance that someone will go to your show, thinking it's the band Aerosmith, and they can take you to court over it. However, if you named your band Jiffy Lube, you're fine, because nobody is going to mistake a band for an oil change.
So basically, there is no way to protect your band name other than proving that someone else of the same name, after you've established your name, is going to be taking revenue from you by confusing the average person.