All the evidence says that the most important factor in hangovers is the way you drink. Drink a beer with a glass of water before the next one and you will be better off regardless of what is in the beer.
Hydration is certainly a part of the issue, but there is significant evidence to suggest that various kinds of toxicity are also at play.
This article suggests that acetaldehyde is partly to blame.
This article talks correlates hangover duration to rates of methanol metabolization. There are tons of other papers out there that identify other potential metabolites.
As for fusels: There is a lot of anecdotal evidence to suggest that beers with high fusel levels lead to headaches, but there isn't much evidence to back it up one way or another.
One paper did explore the issue and even found that fusels lead to
decreased headaches in mice (supposedly), but their sample was very small and I found their methodology a bit suspect. There is enough anecdotal evidence (including my own) correlating fusels and headaches that I'm still somewhat convinced by the link, but I'd don't think there is decisive info one way or the other.
The move to realize that dehydration is an important factor in hangovers certainly appears to be correct, but I would hesitate to say that it is "the most important factor" without some evidence. I've not seen anything that ranked these things, but would love to read anything you might have seen that says otherwise.
Edit: Good find cklages. The bourbon vs. vodka thing is very suggestive. Unfortunately, the decisive research has not yet been carried out, and (thanks to university IRB boards) likely never will be.
Edit2: One thing in favor of hydration, at least, is that it is something you can control. You can't easily make your kidneys or liver metabolize toxins faster, but you can make yourself have a big glass of water before you go to bed.