Why does my car make a noise after driving over a puddle?

This does not sound good for your Integra engine. Here is what may have happened. When you were driving through the puddle water was sucked into the engine through the air cleaner. The water was then drawn into one or more of the cylinders causing what we call "hydro-lock". Let me explain. One of the functions of the pistons is to compress the air/fuel mixture on the compression stroke of the engine. With water in the cylinder the piston cannot complete it's stroke because the water will not compress. When this happens the engine stops. Unfortunately all of the other parts of the engine are continuing to try and turn. When this happens damaged is done to in the area of the cylinder with the water in it. I would guess the noise you are hearing is due to the damage done when the engine "hydro-locked". There is no way to tell the extend of the damage without partially disassembling the engine.

Take out the spark plugs, and spin the engine over. Have someone watch to see if water shoots out of any of the plug holes. If water does come out, and you had tried to start the engine, the water could have bent the valves, or twisted the crankshaft.