Make some white noise on it. Save it as 1.gif in an empty directory
Take it to Imagemagick. Go to Transform and select Roll. If you want snow that falls straight down, put +0+20 in the Parameter box. If you wants snow that blows as it falls, enter +10+20. Save it as 2.gif.
Repeat till you have 5 .gifs in the sequence.
Take the URL of the directory ending in a "/" and enter it into ImageMagick
Note: Some providers don't allow you to link directory without an index page. If our provider is like that, just make a page named .index with this code: 1.gif 2.gif 3.gif 4.gif 5.gif
Go to Output, select .gif in the format menu and in the Delay box, enter 10. If the snow falls to fast for you, back up and enter 20.
Here are some images that are byproducts of making this webpage that you can use.