Chamonix is a small village in France down of the Mt Blanc. From there you can start hiking or skiing in the Alps. But it's also a very good origin for your ride to Italy.
Contents
Hitching Out
South towards Italy
The best spot is located, at a rotary 1.5 miles out of Chamonix. From there every car goes immediately to the Mt Blanc Tunnel so. To get to the rotary, you have to walk the Route Blanches (D 1506) westbound. Take care there, there's no sidewalk, but you can walk on the grass.
Alternative information: Sleep
I found a good and cheap hostel called the 'vagabond' I payed 20E a night and they have free WiFi. The hostel is located near the centre at 365, avenue Ravanel le Rouge (gitevagabond.com) the hostel had a nice international flair and everyone was speaking english.
Transportation
In the case, you should not be able to get a ride thru the tunnel, go back to Chamonix. You can also get a bus thru the tunnel to Courmayeur, Italy (The first village behind the tunnel. Once arrived on the other side, it's easier to hitchhike.
Personal Experiences
I already got a ride thru the tunnel on the 'Route Blanches' just after walking 500 meters. Best is to show your sign with Italy Don't show a specific city, once you're thru the tunnel you can find another good spot near the highway to hitchhike to Milan, Torin or wherever you want to go.