Ten years later, I've come across Jonathan Haidt's book, The Happiness Hypothesis
The first benefit "is that rising to a challenge reveals your hidden abilities, and seeing these abilities changes your self-concept." You can gain confidence from learning that you can persevere even when things are tough. You've seen other people show an inner strength when they were challenged, and you may have said "Wow, I couldn't do that." Only now, you're proving that you can. That's confidence and power.
The second benefit is about how adversity changes your relationship with those around you. Like it or not, your adversity serves as a filter. You find that some people get closer and some get further away. While it may hurt to see it happen, it is easily argued that your life is richer for the kind of people that are still there when the dust settles. From a purely rational, logical, Vulcan sort of way, getting an "improved" list of friends makes for better relationships and can improve your day-to-day experience.
The third benefit is about perspective. "Trauma changes priorities and philosophies toward the present and toward other people." I experienced this in spades. As I've mentioned before, I lost a good friend to cancer a few years after my diagnosis. The combination of MS and that loss helped me not just say "don't sweat the small stuff," I actually live it. It's also changed my attitude and my approach every day. I'm motivated to be more attentive to people, to what my purpose is and to the welfare of strangers.
Please don't get me wrong, my experience has also had some very unpleasant moments. It has also provided some outcomes and growth that I feel very fortunate to have received. Haidt is showing that my experience isn't just mine alone, and it's more than just "seeing the silver lining" stuff, it's real. Regardless of whether you agree with the theory, you should find a way to make adversity work for you, too.