I think it is most likely the wires, rotor cap, and/or coil. When the aforementioned parts get old and tired, then damp with increased humidity, they can leak electricity. This is most easily seen at night, because there will be a nice bright light show under the hood. If it doesn't happen to be raining and the car starts right up, you can test the theory by spraying clean water under the hood just before trying to start the motor. This would not be a deluge of water, just enough to get things damp in the area of the wires, cap, & coil. Also, just because you don't see a light show at night does not mean this isn't exactly the problem. I've seen on a Honda I used to own where the coil arced to the body in a place where it was not easily seen. I finally found the issue, replaced the wires and this fixed the issue.
Something else which is bothering me about the actions you are performing. You said you will pump the gas to get it going. I believe your car should be fuel injected (in fact I'm pretty sure it is). You should not pump the gas in a fuel injected vehicle. This can mess with your tuning as well as not do what you'd expect. There is not accelerator pump on it like there was on carbureted cars.
EDIT: You can actually check it in the dark while it's damp, if you've got the means, like under a carport or inside a garage (door open, please ... don't want nasty Carbon Monoxide buildup!). It won't take long to see what I'm talking about if it's present, well at least for the wires. You can also inspect the cap for any type of carbon tracking, something like is shown in this picture:
Something you can do in the mean time is to spray the inside of the cap and the rotor with WD-40, which will help alleviate the problem somewhat if the problem resides in the cap. The "WD" in WD-40 stands for "Water Displacement", so it will keep the moisture inside the cap down to a minimum and help prevent the electrical leakage.