To the OP: Do you actually want to get out and hike in a white Xmas, or just visit somewhere that will be snowy? By asking about where a "backpacker" can go, did you mean you want to backpack, or just that backpacking is your travel mode and price range?
If you are looking for a community to hang out in with guaranteed snow, you can always go to Minneapolis/St Paul. There is a hostel there,
so that should be reasonably priced accommodations. It's a pleasant place, and likely to be snowy.
Madison has a hostel as well, but we are not guaranteed to have snow. As a matter of fact, they are predicting a very mild winter.
If you actually want to be outdoor/camping/hiking, I would second the recommendations for Bayfield area or the Upper Peninsula of Michigan. The UP can be pretty remote, so a car is a necessity, and that's expensive.
In the Bayfield area, I suggest staying in Washburn, which is waaaaay cheaper than Bayfield - the main tourist center. There is bus service to Bayfield, which is 12 miles away, and the prices are more backpacker-friendly.