I have a snowfall script that some users dislike. I want to have a checkbox to allow users to turn the snowfall on or off. And I want the choice to hold for all pages, and be remembered when users return later.
Can anyone help?

You know when you edit your profile and are given this choice ... View Signatures in Posts?: ... ?
I would make a new option in the edit member profile area by copying that "View Signatures in Posts?:" code as it is applied throughout the forum already. I would rework the copy to cater to your snowfall script. Then I would apply the new variable where ever you present the snowfall script in your site.

I considered that, but I was really hoping to avoid modifying the database.

Well, do a search through the code for Response.Cookies to see how cookies are done and then do a duplicate of that code.