How often do kids get to play in the rain without getting yelled at to get inside? If there's no lightning, let them slide around in the mud for a while. A water balloon fight could help clean them off. Turn on the sprinkler in the rain and tell them they get to run through water from every direction.
When you move the party inside, have a human ring toss contest. Have two big inflatable pool rings, and have the kids get into two lines. The first person in each line takes a few (5-10?) big steps away and stands with their hands at their sides. The second person throws the ring and tries to get it to go around them. Then, that person goes and stands with their hands at their sides, and the next person throws, etc.
If you set up a tent inside, the kids will love that, even if there's nothing inside it.
Play "pin the crown on the mermaid."
Get a jump rope and have the kids get in line to play "high water, low water." If I remember the rules correctly from my own childhood, I think that one kid holds each end of the rope, and they hold it at varying heights, slowly moving it up and down or jiggling it. Everyone chants "high water, low water," and the kids take turns jumping over the "low water" or diving under the "high water."
Serve lemonade with fruit and food coloring to stir in (just a drop will do it). Pink lemonade is cool, but blue lemonade is cooler, and blue lemonade with frozen raspberries instead of ice cubes is the coolest ever.
Let the kids made "dirt" and "sand." Give each kid a clear plastic cup and a spoon ("shovel"), and put out bowls of vanilla pudding, chocolate pudding, crushed up oreos, crushed up vanilla wafers, and gummy worms. Let them layer it however they want.
You can pour some blue jello into cups, let it set most of the way, add gummy fish and sharks, and them pour in more jello and let it chill until it's done. You'll have little aquariums.
Play "musical beach towels" instead of musical chairs.