The stage names of steampunk band The Extraordinary Contraptions: Aelus von Stadberg, Dimitri von Stadberg, Sephora Bostwick, and Corbin Welch. (Their real names are Alex Stadler, David Tyberg, Sharon Burdick, and Kevin Guthrie, respectively).
Singer/songwriter Eagle-Eye Cherry. That is in fact his birth name: He got it because when he was a baby, he looked at his father for the first time with only one eye open. The name can cause a sort of inversion of I Am Not Shazam, since "Eagle-Eye Cherry" just sounds more like a band name than it does a solo artist.
Heavy metal guitarist Guy Mann-Dude, though his birth name is Guy Shiffman.
Guns N' Roses pretty much thrives on this trope. The band-name itself is derived from the Stage Names of founders, Tracii Guns and Axl Rose. The classic Line-up had Izzy Stradlin, Duff "Rose" McKagan and Slash. The latter two and Axl are still in the band along with Sixth Ranger Dizzy Reed, and late additions (back when it was Axl, Dizzy and some guys) Richard Fortus, Frank "Thunderchucker" Ferrer and Melissa Reese. Past members include: Buckethead, Ron "Bumblefoot" Thal", DJ Ashba (DJ being short for "Daren Jay". He's not a DJ as such), Robin Finck (from Nine Inch Nails) Tommy Stinson, and Chris "Mother Goose" Pitman. It seems that having an awesome name is just as important as being a talented musician, if you want to join Guns N' Roses.
Joe Satriani. Doesn't really matter if it's his given name or not, it's still a pretty awesome last name.
Deep Purple has has a few members with cool names through the years (as is to be expected with 8 different lineups, not counting the times they've reverted to previous lineups). Some noteworthy members are Ritchie Blackmore, Steve Morse and the late Jon Lord.
Australian singer-songwriter James Keogh already has a decently cool-sounding name. But take the name of an actual Cool Old Guy from an Australian novel and become VanceJoy, and it instantly becomes even cooler.
The Mechanisms: Stage names they may be, but the band all perform in-character and thus answer to Jonny d'Ville, Ashes O'Reilly, the Toy Soldier, Drumbot Brian, Gunpowder Tim, Raphaella la Cognizi, Marius von Raum, Nastya Rasputina and Ivy Alexandria during their gigs and on their albums.
For a guy with a stage name as mundane as "Captain", Captain & Tennille's male lead was masking a rather fantastical real name: Daryl Dragon.
Iron Maiden has guitarist Janick Robert Gers and drummer Michael "Nicko" McBrain.
Neonfly guitarist and founder Frederick Thunder. No word as to whether or not it's a stage name just yet.
Influential jazz drummer Art Blakey.
An in-universe example from band Glory Hammer, On their first album Tales from the Kingdom of Fife the hero is "Angus McFife". On the second album Space 1992: Rise of the Chaos Wizards which is set ten centuries later, his descendant is called "Angus McFife XIII".