There was a time in New Orleans where he looked like he was on his way to being a truly dominant, consistently Top 5 NBA Player; someone who could be one of the 20 best NBA Players of all-time. He came into the league with tremendous hype; a dominant season in college, followed by an Olympic run. Everything about him screamed next super-duper star. But he has never been that player; he puts up good numbers but he's inconsistent and doesn't seemed wired like the kind of player who is the best player on multiple championship teams.
I will say that the original idea for the Lakers was that as LeBron got older, Davis would step in and be the fulcrum for the Lakers as he is in his prime, as
@Euclis20 points out. But to give Davis a bit of an excuse, that really isn't the case. LeBron is still the dominant player, the team was built by GM LeBron and designed around him with old players that LeBron thinks are still useful guys. LeBron is the dominant ball-handler, plays the most minutes and controls the game. It's hard for Davis to assume the mantle when LeBron clearly has no interest in relinquishing it.
LeBron's role on the Lakers is pretty fascinating. Statistically, he's having an awesome season considering his age, and it's hard to look at his numbers and think that he is in anyway responsible for the Lakers being bad this season. At the same time, it's a team designed basically so LeBron will have to do it all and at this stage of his career "LeBron doing it all" isn't necessarily as conducive to winning as it once was.