Every season has at least a couple of events you can participate in for fun. If you win the ice fishing challenge in the Festival of Ice, you can earn 2000g. It takes place on the 8th of Winter. Then there's the Night Market, as mentioned above.
Even if you miss these two winter events, there's still the Winter Feast in Stardew Valley. It happens on the 25th between 9am and 2pm. Mayor Lewis will send you a mail telling you who you'll be playing secret Santa to on the 18th. The item you give the gift recipient at the Feast of the Winter Star will be worth 5x the normal amount of friendship points.