Toy Boat Races
If natural pools of water are forming in the yard, use this as an opportunity to race toy boats. Kids can put toy boats into the puddles and watch them sail from one end to the other, or twirl around and around in a circle. If you don't have toy boats to sail, use rubber duckies or other bath time toys that are buoyant.
Mud Men
Instead of building snowmen in the snow, build mud men in the rain. Playing in the mud should be done in clothes that are okay to get dirty, because it's possible that some stains won't be able to come out. To build mud men, start with a large hump of mud and gradually build up from there. You can use twigs for the mud man's hands, tree branches for the arms and a carrot for the nose. (You might not, however, want to wrap your favorite scarf around mister mud man's neck.) You will be more successful making a mud man when the mud is thicker and not too runny.
Human Rainbows
If you have washable crayons or paint laying around, put on a bathing suit, paint your body and go out into the rain to become a human rainbow. Bring the crayons out there with you so that you can keep coloring on yourself, or on someone else, while the rain helps wash old colors away. Go inside to dry off and then repaint yourselves for a blank canvas. If someone has a waterproof camera handy, this is an optimal photo opportunity.
Water Harvesting
If you are interested in finding out how much water you can collect from the rain -- a process known in the environmental industry as water harvesting -- set out beach buckets and pails around the yard and let the the rain water accumulate inside of them. Keep an eye on the buckets and bring them inside when they fill up. Now that you have fresh water you can choose what to do with it, such as store it or use it to water indoor plants. Never drink the rainwater, however, as it may not be safe for ingesting.