Hand-eye coordination and timing are two benefits of speed bag training, although like someone else said you can get that from the rest of boxing training. The speed bag is just another way of working it, adding variety into your training routine, which is always important. As you get to smaller bag sizes, it tests your accuracy, timing, and coordination. It's challenging and forces you to improve yourself in these areas.
The speed bag I have isn't particularly fast, so I leave it slightly less inflated than a normal speed bag. This forces you to hit it harder. Sometimes I'll just spend a week training my jab; shadowboxing with weights, on the heavy bag, on the double end bag, with mixed up footwork, and on the speedbag. Spend a couple rounds of three minutes hitting the speed bag with your left hand only. Your shoulder and tricep BURN, and I swear it improves my jab like nothing else. It strengthens and works the shoulder and tricep, which are the crucial muscles in the jab, seeing as how it's not a power punch.
The other way I use it is to try to go as fast as I can with my hands. Instead of letting it bounce three times (strike-bounce-rebound-rebound-strike-bounce-rebound-rebound...), I strike it after the first rebound (strike-bounce-strike-bounce), alternating hands with each bounce. I try to hit it harder, so the bag goes faster, which forces me to make my hands move faster. I'm sure there's another name for them, but I like to call them "Speed Bag burnouts." I'm first trying to go as hard and as fast as I can, and then I try to do it for as long as possible. This, to me, feels almost like I'm just injecting lactic acid into my arms. They turn into rubber in no time.