H.W.
This is an alternative suggestion, and take it with a grain of salt if that suits you.
I run a preschool and we go outside every day. Unless it's 100% dumping from the sky, we are out. Consider investing in rain boots, rain pants, a rain jacket, a fleece jacket for underneath, and warm socks (and possibly a warm hat to put on under hood.) Then duplicate this ensemble for yourself, and get outside to the park or wherever it's fun to run. Dr. Spock (the old-school pediatrician, not Mr. Spock from Star Trek, ha ha) said that kids need about an hour of time outside every day. After working with kids for over 19 years, I totally agree with this. Of course, if it's very rainy, send them out for a shorter time, but it's good for kids to go out and run in the rain, stomp in puddles, even if it's a neighborhood walk for a half-hour. You'll both come in refreshed and ready to be indoors, and the fresh air and puddle-stomping will do your son a world of good!
For what it's worth, my son's preschool, and a few others I know of, do this too. In Europe, there are "outdoor schools" in which the preschool children are outside for most of the day, unless the weather is inclement. Years ago, I nannied two families of four girls, and we were outdoors in all weather. The day I heard them singing "We are Oregon Girls!" as they splashed in puddles was a wonderful one. We have to raise hale and hearty creatures in our soggy town!:)