In reply to Mr Spoons:
I understand the above mentioned Les Nigritelles to be an excellent option though have never stayed there. If you prefer to be in the throbbing heart of the metropolis that is Cogne, so as not to miss the bright lights and splendour, Residence Chateau Royal provides centrally located apartments, with breakfast if you want it. Same people own the Hotel La Barme up in Valnontey, a favourite with many. Going upmarket Residence Maison Pierrot, just down the road from Chateau Royal is a good choice.
In La Grave you could do worse than Gite le Rocher, though I'm sure there are many options.