I briefly looked a bit into this when Porsche put KERS into the GT3R and quickly became disappointed.
As Geoff said above, look at how often you actually use the brakes on your bike, it probably isn't much. Even on moderate downhills, you usually just maintain speed for the inevitable uphill that follows probably 95% of the time. Sometimes you have to stop at the bottom of a hill and that's no fun, I admit.
If you figure that you want something like enough energy for 50' of climbing for something like 200 lbs of rider and bike, you need around 13.5kJ stored equivalent to that same rider going 38 mph. Figure that if you have to store that, and something goes wrong things are going to hurt a lot... Imagine being hit by a cyclist moving at near 40 miles per hour...
If you want to store that energy in a flywheel, it will need to really be moving if you want to keep the weight somewhat reasonable. Probably going to need to keep it a vacuum to prevent unnecessary losses, the bearings need to be very good look out for any little bumps to damage things... things get complicated very quickly.
Batteries would probably be more reasonable and tend to fail less spectacularly (though still quite spectacularly) but you run into the other issue of you probably don't actually use the brakes that much. Most of the work at any reasonable speed to to push air out of the way and other general losses. when you aren't pushing air out of the way, you aren't going very fast and don't have much energy to store anyway.
You are probably just better off with a lighter bike if you do a lot of stops and starts...