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.
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.
Aug 2018: Could not find a good spot near the rotary, but found an ideal lay by a few hair pins up the road towards the tunnel, right by a toilet. Approximate location: 45°54'18.1"N 6°51'38.8"E. Forest trails lead up to this point (don't walk on the highway). The pull over spot is soon after a hair pin so most cars will have plenty of time to see you. A sign is recommended, as you probably want a ride that takes you in the direction you want (Milan or Turin/Genova), rather than to Aoste.
Free: There's a laundry service room somewhere in the main shopping street, open 24/7 even with power. Look for that and stay the night there. If you wanna hitch the tunnel be aware of a lot of police. They didn't say anything through because of the hitchhiking on the speedway just after the roundabout. But it's still better to go up in front of the tunnel, better to hitch from there. Rue du Dr Paccard is the road called. It's in a little Mall I think on the bottom floor a bit hidden behind some stairs. Must be also limit WiFi there It's on the northwest side of the road, away from the river. Good luck.
Otherwise: 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.
In the case, you should not be able to get a ride through the tunnel, go back to Chamonix. You can also get a bus through the tunnel to Courmayeur, Italy (The first village behind the tunnel. Once arrived on the other side, it's easier to hitchhike.
I already got a ride through 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 through the tunnel you can find another good spot near the motorway to hitchhike to Milan, Torino or wherever you want to go.