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If you're determined to run on a rainy day, read the labels and look for tights or pants that are breathable. A 100% nylon pant or polyester blends will certainly keep you dry, but they can be too hot for intense activity if it is rainy but not particularly cold. But the nylon and polyester blends are fine for warm-ups, cool-downs, and casual running around after the kids in the yard.
If the weather is wet, but not cold, most moisture-wicking pants, tights, and shorts will keep your skin reasonably dry. The pants will get wet and absorb some water, and your skin may feel damp, but unless you're out in the rain for hours, any moisture-wicking pants and shorts will keep you from feel cold and clammy.