La Résistance and the people of Zion from The Matrix tend toward this, mainly to separate themselves from their former identities in the Matrix; having presumably chosen the names themselves at least partially justifies the trope. Even some native-born residents of Zion sport handles for names, like Tank and Dozer.
When Mr. Furious is asked his real name, he mumbles out "Phoenix... Dark... Dirk" before finally admitting it's really just "Roy."
The villain's name is equal parts cool and silly: Casanova Frankenstein. This isn't a name he assumes after becoming evil either.
In the RiffTrax treatment of Daredevil, when Elektra initially refuses to tell Matt Murdoch her name, Mike Nelson fills in for her: "The name's Frigid McTurndown." Then, when Matt and Elektra start sparring, Bill Corbett adds, "Okay, I lied. The name's Stompy Von Asskick." When the character does reveal her full name (Elektra Natchios), they add "Electric Nachos?" Just can't win.
In Star Wars: Luke's original surname was "Starkiller", but after a reworking of the character it was changed to Skywalker, which most people think is still pretty cool. Starkiller was later given to the player's character in the Force Unleashed video game. Many other characters have made-up names that some people find cool. Mace Windu, Han Solo, Lando Calrissian and Darth Vader are a few of the most popular examples. Also "Death Star", "Star Destroyer", "Lightsaber", ...
MST3K's riff on Space Mutiny has an absolute firestorm of these as nicknames for the hero, such as Gristle McThornbody and Big McLargeHuge. So many that a few of the others on this page are direct Shout Outs. The funniest is the last one; Bob Johnson.
Ray Allen's character Jesus Shuttlesworth from He Got Game.
White Goodman: Allow me the pleasure of introducing you to Blade... Laser... Blazer...
Derek Zoolander. Whose name is infact a parody of real-life male model Mark Vanderloo. The original title for the film was actually Zanderloo, before flipping it around. Vanderloo is incidentally the facial model for the default male Commander Shepard from Mass Effect.
The Australian short film The House of Names revolves around two protagonists who spend all their time trying to think up something cool to change their names to.
In Boogie Nights the main character changes his name to Dirk Diggler. That's just okay, but the name of his character (in the Kung Fu ... er ... something-em-up he wrote) is named Brock Landers. (His partner Reed Rothchild's character is the even better Chest Rockwell.)
The B-scifi movie Trancers, and its many sequels, revolve around a time-travelling future cop by the name of Jack Deth. In a subversion, everyone he meets outside of his own time period mocks his name relentlessly.
The eponymous protagonist of Major Payne. His full name is Benson Winifred Payne, which isn't too great at all, but once he's joined the Marine Corps and attained the rank of Major...
Black Dynamite. Played for Laughs, as this is apparently his real name. In one scene, a few neighborhood girls remark that both their mothers told them that "Black Dynamite" is the name of their father. Looking guilty, our hero asserts, "A lot of cats have that name..."
In Guillermo del Toro's Pacific Rim, character names such as Yancy Becket, Hermann Gotlieb, and Tendo Choi. Idris Elba plays Marshal Stacker Pentecost. Ron Perlman is Hannibal Chau (Invoked Trope; it's a mash-up of his favorite historical figure and his second favorite restaurant, he just picked the name because it sounded cool). Charlie Day is Newton Geizler. Charlie Hunnam is Raleigh Becket. The older Australian pilot is Hercules Hanson.
From Gladiator, the protagonist's name is Maximus Decimus Meridius. Sounds like a serious individual and he is. The writers basically ignored Roman naming convention in creating it.
Han Solo Berger in That's My Boy. No one can figure out why he uses the name Todd Peterson for his profession.
One of the characters of Giant is named Jett Rink, which is made even cooler because he's played by James Dean (in his final role).