Take a look at the character's abilities. Does anything special about the character's powers help provide input for a name? For example, the Marvel comic book character Magneto received that named because of his abilities to control electromagnetic energy.
Look at the physical qualities of the character. Is the character monstrous like the wooly X-Man character Beast? Is the character muscular and ogre-like like The Hulk?
Be inspired by the character's costume. You want the entire character to be a cohesive unit. She should be easily identifiable to an audience, so the name and the wardrobe should match. A perfect example of this synthesis is the DC comic book character Batman. Although he doesn't necessarily have bat-like abilities, he dresses in a bat-like fashion.
Look at the character's history. What made him a superhero? Is there a tragic past? Perhaps your character gained his abilities through an accident with radiation or perhaps the character has a genetic mutation that provides special powers. For example, Spider-Man gained his abilities by being bitten by a radioactive spider.
Write down all the names you would consider for your character.
Go through each name, and try to determine which has the most sticking power. In other words, see what name stays in people's minds the most. You may want to use friends as test subjects. Read them your list of names, and see which one they recall the easiest.
Pick a name for your character. Once again, go to your friends and colleagues, and ask what they think of the name. Take their opinions into consideration. In the end, it's your decision, but if more than one person shows disapproval or brings up the same critique, that may mean a large number of the public may feel the same way.