First birthday dilemma
I am going to assume family and friends home would be a no go?
A park is a great idea, but if the weather is foul as you said.... :(
Would maybe a day out at a zoo/skating rink/something like that where every one pays their own admission but you all do something fun together work?
We often do things like that for family get togethers and parties. It keeps the cost down, doesn't overwhelm the kids as much, and makes a happy memory for all.
Hopefully some one will come along with some better ideas for you!
Celebrate it in your home. It's your child's first birthday party and it will going to be memorable if it you'll celebrate it in a place where he blossomed, and it's in your home! You can set it up in your garden if you have. Or find an empty lot in your area, near your home and ask permission if you can have it there.