We faced this with my daughter's birthday party. We had invited all of her kindergarten classmates, plus a few old friends, and I had a feeling it might rain and wash out our planned picnic in the park, so I put down our house as the rainy day location. It poured and stormed to beat the band that day!
I moved our pinata to the garage, took the huge baskets of water balloons to the garage, and had a lot of the party in there. Inside my house, we had a pin the tail on the mermaid game and we did quiet guessing games. And oh yes, kids ran up and down my stairs and all around. My house is quite small, and it was a zoo, but I had two older girls helping me and all of the parents felt sorry for me and kept things semi-controlled. I stood outside in the pouring rain with an umbrella cooking hot dogs and bratwurst on my charcoal grill (which lighted beautifully in spite of the rain!). Months later, I still have kids and parents telling me it was a great party. When you get right down to it, all kids want to do is play and have fun, and they'll find a way to do that. It's just keeping them corralled and safe that is the issue, and as I said, the other parents and my helpers were great about that. Once we ate our lunch and had cake, they were really all quite calm.