A swarm of disposable quadrotors surrounding a ship, set on a rotating recharge schedule, that intercept missiles and detonate a small explosive payload.
There are destroyers on the drawing board with enough excess capacity to power rail guns, and the latest aircraft carriers have upgraded reactor units to power the electromagnetic launch systems they use instead of steam catapults.
The reactors that supply these large ships should have plenty of excess power for a few hundred drones.
>These things might be too slow to intercept an incoming missile, especially the super sonic ones.
They don't have to match speed, they just need to place themselves in the path of the missile; they are incredibly agile and shouldn't have trouble doing so.
You have to place Phalanx systems all around the ship in order to provide 360 degree protection. However, an enemy could concentrate multiple missiles on a space covered by only 1 Phalanx.
If radar detects multiple missiles inbound, more drones could be easily be routed to the appropriate area to intercept. To overwhelm the defense you'd have to fire more missiles than the ship has drones (and since drones could be made much more cheaply than supersonic missiles, that would be fairly hard to do).