Not every album has a winter song at the ready and not every band can fit the mood. Oh sure, there will be the odd Christmas song if you can stomach them, but Christmas is only a twinkle in the eye of winter. It doesn't define the season, rather it is often the other way around. But if we turn to games, we can find many, many soundtracks that address a heavy snowfall on a cloudy night.
Now, I could spend my time compiling a list of top tens and twenties to teach the lil' ones all about the classics and how great this snow level's theme or that snow level's theme is, but I'd feel a lot more satisfied to bring up just one good song: Donkey Kong Country's Ice Cave Chant.
What begins as slow intense build, almost ominous, is soon met with a sequence of notes in fast succession. On one hand, the listeners feel alerted toward some impending hazard. On the other, they feel excited. Then comes the lighthearted, xylophone tapping and a child-like joy in the music. This is only occasionally interrupted, so the fast pace, repeating notes can linger in the mind, reaffirming the mystery and excitement.
I'm...clearly...no music theorist, but I understand mood. And the one created in that song is a mixed one, like the snows of winter themselves. The first heavy snow is exciting and beautiful. It brings people back to their first snow, whether it be as a child or an adult, and it fills them with wonder. Snowballs and snowmen, the snow is for laughter and playing! Yet, it can be dangerous. It's slippery and cold. It's apparition white and obscures vision, making it as mysterious as it is inviting.
Snow encourages all those feelings, and that one song encapsulates them as well. The two go perfectly hand in hand. While I'd wager there are a number of video game tracks that can service this emotional state equally well. Ice Cave Chant is the first that comes to mind and the one that's stuck the longest.