I'd suggest Les Diabrelets or Champery in Switzerland. Both have hourly rail service in the morning to GVA, 2 1/2 hour travel time with a single change at Aigle. By "after New Years", I assume you know it may be difficult to find lodging anywhere in the Alps for just a few days. Most want a one week minimum until the New years holiday is over, which probably will be OK in 2015 with New Years Day falling on a Thursday.
"Winter activities"? I assume you mean downhill skiing or snowboarding. From Champery you can ski into France if there is enough snow for the connections to be open. There will also be sledding and cross country skiing available. "Not too touristy"? These are not big name Alpine destinations like Verbier, Zermatt or the Jungfrau region, but hey, skiers are tourists too and there is a level of services to handle them.