UKH

Trying to get to Courmayeur early on Wednesday morning to walk up to the Gonella Hut. Heard we need to start this walk at 5/6am because of the glacier conditions. Is there a cheaper way of getting round to Courmayeur for an early start from Chamonix without taking a taxi?

Try BlaBlaCar - it's not public transport, but would be cheaper than a taxi.

Hitching or go the day before?

You can book a coach from cham bus station to courmayeur. Really cheap and regular. 

From memory I don't think the earliest coach will get you through the tunnel for 5/6am.

I've hitched through the tunnel a few times, even with my heavily loaded touring bike. Perhaps you could advertise that you would pay part / all the tunnel toll,  that will surely interest some impecunious soul !