To celebrate the beginning of the new year, the residents of Mineral Town like to hold festivals. There are two different events you can attend with them, but you can only go to one of them.
The first one is a New Years Feast gathering at Rose Plaza. Mayor Thomas will let you in at 6:00 pm. There is a large table full of food. Yummy! Talk to Thomas from the opposite side of the table to start the event. You'll be fed some New Years Tempura. At the end of the meal the Mayor will fill the empty slots of your backpack full of New Year Buckwheat Noodles. They're different than the Noodles that Jeff sells at his shop.
The other event is on the summit of Mother's Hill. Around midnight head up to the top of the mountain to watch the sun rise on the first day of the new year. Some of the other villagers will be there too. To get this event started talk to everybody and then sit and watch the new day. You don't receive any gifts from attending this event.