I bake an assortment of cookies every December for food gifts and for us. I usually split this into two, one earlier in the month and one closer to Christmas. I do the food gifts earlier before people might bake for themselves and then ours later, closer to Christmas. I typically bake the same things but may add in something new to try. So far I haven't found anything new that has been good enough to add to the annual rotation that I can remember anyway. I typically stick to sugar cookies, gingerbread cookies, peanut blossoms, peanut butter balls, oreo balls, peanut butter fudge, sometimes peppermint bark for gifts, snickerdoodles....I think there is more but I can't remember at the moment.
Since we don't really get cold here in the winter I don't typically do any special foods in the month of December as a whole aside from the sweets.