I wrote about just this question about a month before the invasion was actually launched. This was after reporting on the story, and shifting my own ideas back and forth a couple times, over a period of about eighteen months. There were so many different rationales, each distinct, often contradictory, one interchanging with another as the news cycle changed, that it eventually seemed clear to me that none of these were really reasons. They were rationales - arguments you devise (convincing or not) to make the case for something you want to do, rather than the actual motivating force behind your desire. So why did they want to do it? At some level I think it had simply become an idee fixe for many of these people. Because for many of them, when I would have frank conversations with them, they had a difficult time getting past the rationales, even in what I think were off-the-record and unguarded conversations. The real underlying reason, to the extent there was one, was the notion of creating a transformative event, a democratizing wave in the region that would get away from managing and on to 'solving' deep and lingering obstacles to American power.And, frankly, that is the worst of all possible reasons. The US was already involved in Middle East politics because of its strategic and economic interest in oil and, to an extent, because of its relationship with Israel. The US is still heavily involved there for those reasons and this will continue to be the case until long after all of us are dead and gone.
But the war happened because some old hands at running the show over there were bored with playing the game. They wanted a transformative event to reset the pieces. Maybe they'd get lucky and the resulting chaos really would "solve" some problems they'd grown weary of "managing." There wasn't any way to know how that would go but that wasn't really the point.
The calculation was this. We've been futzing around here forever. We're bored. Let's do something different. The terrible cost in blood and treasure of doing something different didn't really matter to those deciders. They ended up getting richer anyway.