I'm a pretty good armwrestler, I can beat people with some far superior lifts (such as the bench press, curls, pulls).
In my experience, unless you're really shoulder dominant in the bench press, its pretty worthless when it comes to armwrestling. I've beat people who could almost double my bench press when I was younger (note that they were also pretty much one trick ponies, haha).
One key, which someone already stated, is armwrestling ALOT. I don't armwrestle much anymore (just whenever I find a good challenge), but when I was younger, I would armwrestle my cousin every chance I got.
Like any other lift, once your body gets comfortable with that motion, your strength is going to increase by a good amount (think about your first few weeks of bench pressing).
I think shoulders are definitely underrated in armwrestling, which is strange since I believe its the most important muscle in the movement.
Wrist strength would come second because if you can dominate the wrist, not only does it give you better leverage, if the other person's wrist is at a weird angle, it weakens the rest of the arm. Listen to shizen and get a sledge hammer, this site has a few great exercises for it...
You could also try making a wrist roller. The wrist can really make the difference between winning and losing.
I'm not trying to bash steel nation's post, but I've never found reaction time to have any effect in determining the outcome. It might appear that way only because the stronger person is going to get a better start.