In reply to Nath93:
What's best to do in Coumayeur in January strongly depends from the weather. If there's a lot of snow, the best thing is probably to ski the slopes - between Courmayeur and La Thuile there's plenty of technical things to do.
If the snow is there but it's not too much, a popular activity in the area is snowshoeing. There are many interesting hikes you can do in Val Ferret and Val Veny using snowshoes, inevitably panoramic and not too difficult.
If the weather is cold but there's no snow, go ice climbing. There's dozen of lines you can reach driving less than 30 minutes from the village. Ask to the local guides bureau for directions.
If you'll be around, send me a message, I may be in Courmayeur too