Edmund is Peter's younger brother, a second king of Narnia, and a hero in his own right. He may not be as strong as Peter, but he's crowned Edmund the Just and is still remembered for his wisdom and knowledge. We see some of that wisdom come to the forefront in Prince Caspian, since it's Edmund who sets their course for Caspian's camp. But mostly, we get to see how Edmund has finally—
We don't get much from Edmund in Prince Caspian that we didn't see in The Lion, the Witch, and the Wardrobe. Instead, his character development from that novel follows through into this one. How's this for an example? When the others don't see Aslan but Lucy does, Edmund puts his faith in Lucy. While he didn't see Aslan personally, he does say:
"When we first discovered Narnia a year ago—or a thousand years ago, whichever it is—it was Lucy who discovered it first and none of us would believe her. I was the worst of the lot, I know. Yet she was right after all. Wouldn't it be fair to believe her this time?" (9.81)
Clearly Edmund has learned his lesson and is willing to put his faith into things he cannot see. As a result, he's the second Pevensie to see Aslan again after their return to Narnia. He also has faith in his big bro; he believes that Peter knows what he's doing in the duel, despite the uncertainty of the outcome.
But wait! Edmund still has a bit of Edmund in him, so it's not a complete 180. When Trumpkin suggests they won't be of any help in the war, Edmund gets "red in the face" (8.19). Peter tells him not to lose his temper and to suit up in armor, but Edmund argues that he "'[does]n't quite see the point.'" Ouch. Eventually, though, Edmund relents and follows his brother's orders, getting a chance to challenge Trumpkin to a duel and whollop him one.
So there's still a little of the old Edmund boiling beneath the surface. But in Prince Caspian, we see how Edmund has tempered that side of his personality with the lessons he previously learned in The Lion.
That's what sequels are for, right?