Having a picnic inside is a strange enough concept that it may make your kids give you funny looks when you announce it, but if they're just the right age, they'll go along with it. This is not an occasion for fried chicken and apple pie, but may be worth hard-boiled eggs and absolutely calls for cutting the crusts off the bread if that's how they like their sandwiches. Make a pitcher of lemonade - not a thermos of hot chocolate. Spread a tablecloth or a beach towel on the floor. If you usually use a picnic basket for picnics in the summer, then use it. (Be strange, but not too strange.) Use the tacky plastic summer cups and plates.
Surround your picnic site with house plants - just use whatever you have around. Have something on the sound system - my mother used to use a vinyl record (remember those?) of bird calls, or Tijuana Brass, the one with tropical sounds in the background. Maybe steel drum music will keep their little feet tapping. I'm sure nowadays you can download birdsongs for free onto your MP3 player, and then broadcast it through the house. Maybe even perfume the air?
And here's where everyone starts to giggle. (But you, Dear Reader, must maintain the suspension of disbelief.) The checkered tablecloth is on the floor. (You are sitting on the floor, too, of course.) You have the music of bird song field guides playing (without narration, if possible). If you spring for an entire CD of bird song, you will totally convince your small diners that they are not, in fact, on the floor eating tuna fish, but Somewhere Else entirely with the lush tropical flora of Florida (or is it still the living room?) in the background. You're all munching away on your egg salad discussing the weather ("is this the nicest spring day in history, or what?"). Never mind if it's cold and gray out the window.
Pack for all contingencies. You may find you need sunscreen and bug spray as well - it may not keep away the mosquitos but it ought to be good for a laugh. Sunglasses and hats may help keep the sun out, even if there isn't any. Consider binoculars for bird-watching - migratory birds as well as equatorial species often are spotted near the heating vents!
[This is also a fun birthday party idea for little children. Have everyone's celery and carrot sticks along with a half a peanut-butter and jelly sandwich, a juice box, and a party noisemaker in a brown paper lunch bag.]
So don't let the weather OUTSIDE determine what you do INSIDE! Here's the next answer to "what can I do-o-o-o-o?" Pack up the picnic basket, collect your children, and set off for the great indoors.