One of the mainstay features of DC Comics is the so-called "crises". To those not familiar, this is effectively editorial hitting the reset button on the universe, allowing for simplification of the existing continuity.
While reboots are all the rage in Hollywood these days, DC was far ahead of their time when it came to re-inventing their heroes. As a result, the company ended up with a lot of what are now referred to as legacy heroes - superheroes that chose to take over the name of another character that has gone before them.
In this way, along with radical changes due to continuity resets, complete overhauls of origins, power sets, and secret identities are not uncommon in DC. Their competition, most specifically Marvel, on the other hand, tend to embrace the wackiness of some of their older heroes. Sure, most major superheroes have a whole list of villains that will never appear in a major motion picture, and dumb costume changes that would embarrass Liberace, but Marvel chooses not to tamper with their history. They stare at the evidence of their awkward youth head on, whereas DC burns the pictures, and avoids certain neighborhoods entirely (you never know who you'll run into).
These are the characters that managed to somehow grow into their cool selves, despite an awkward decade or two. Many of the heroes on this list are simply products of their time. Some are better known, and others are a little more obscure. What unites them all is that, while looking back at their origins may not be advisable, they are now well-respected heroes in their own right.