There are two museums side by side on Chambers street - its the old victorian bit which has been closed for three years. The modern half which is also the Museum of Scotland is still open and lots of fun for kids. It's free, its big - loads of things to see, dressing up and activity bits. I'd highly recommend it.
You could also try the museum of childhood on the High Street.
If the weather is good (yeah - I know. But hey, miracles might happen!) you could climb the Scott Monument. Great view and good twisty staircase inside. Not great if you have vertigo though.
Or the Camera Obscura?
Castle is interesting but V expensive I believe.