01 tiburon runs rough after rain

Check for water getting into electrical compnents.

When the light is on have it checked for DTC'S - diagnostic trouble codes , do you know what they are ? Ignition system problem , coil's , plug wires if the vehicle has then . The check engine light code is probably for a engine misfire . Check Engine Light On

very common problem with your year model is missing inner fender not sure why but if one is missing good chance when it rains u are soaking the coil causeing it to arch and miss fire or arching at connection at the plugs and wires if ur missing a inner fender dont waste ur time replaceing plugs, wires, or coil tell u have replaced the inner fender

spark plug wires need to be replace

replace the spark plug wires

Sounds like a secondary ignition problem to me. See if you can duplicate the symptom by using a spray bottle to spray a "mist" around the engine compartment.

Water is probably leaking onto your engine somehow and shorting out the plugs.

I just had this exact problem. Wouldn't start or took 15 trys to start, but then ran smoothly. Sometimes a little rough starting on dry days, but pretty much hard starting after rain. I was thinking water in my tank/bad gas, but shooting some starter fluid in my intake didn't give me the initial blast at start up... that certainly indicated that it was electrical...
...but the next day the vehicle stalled in the middle lane of a three lane street, wouldn't restart. After pushing it to the curb, I got to the point where I was checking the wire from the coil to the distributor cap. Two thirds of the wire lifted right out. This wire was in 2 pieces! Amazing that it had been running at all, with this one wire burned down to a whisper, before it broke completely. There were signs of heat/elec damage to the wire also at a point an inch away on this wire.
Replacing the wires is absolutely the first thing to try. It's cheap and easy to do, and it should be done (or the wires inspected/tested) routinely anyway. The coil-to-distributor coil gets more engine heat and more use than any of the indiv spark plug wires and hence wears much faster.
I am betting this wire is much more likely your problem than a coil failure. Good Luck!

replace distribudor cap

It sounds as thou you have moisture in your distributor.Pull the distributor cap off and let the car sit and dry out for a couple of days and see if that fixes the problem.