Mitzpe Ramon is a city in the Negev region of Israel. Highway 40 is the sole highway that runs through the down, which runs from Beer Sheva to Eilat. The town is located on a cliff overlooking the Ramon crater. The town has a vibrant artistic scene, with many cool DIY bars and artist hangouts.
South towards Eilat
Bear in mind that Route 40 is the least busiest route between the north and Eilat, so hitching south might take some time. Also note that there virtually no settlements in between here and Eilat, so accepting a lift with someone going part of the way is not recommended unless they're going to drop you off at Highway 90. Such conversations might be tricky if the driver doesn't speak English (something that's quite common in rural Israel).
The best place to get a lift is just after the second circle (southern-most of the two circles) on the road just before it wraps around the corner and goes down the cliff. Standing at the petrol station might be frustrating because (1) a lot of traffic passing there is still only ending in Mitzpe Ramon, entering the town at the second circle, and (2) you'll miss a lot of the traffic leaving Mitzpe Ramon that leaves at the second circle. A good place is here: https://goo.gl/maps/SeFLZTcM44BjTVyv5
After being dropped off by his ride halfway between Mitzpe Ramon and Eilat, User Ryan got a lift at the T-junction of Highway 40 and Highway 12 near Neot Samadar by standing just after the restaurant here: https://goo.gl/maps/hN9VPWdSAvnpUNURA
North towards Sde Boker, Beer Sheva
After Sde Boker, Beer Sheva is the next major city. Traffic is plentiful, and getting a ride won't take long. To skip local traffic, it's best to walk to the Dor-Alon service station. The vast majority of cars travelling north will be going in the direction of Beer Sheva.
East towards Arad or Dead Sea
For this trip be prepared for a multiple rides. You will first and foremost need to travel north to Sde Boker. Drivers on their way to Beer Sheva will be able to take you there. At Sde Boker, ask them to drop you at the junction between Highway 40 and route 204, where you can start hitching for Yeruham/Dimona, and from there to Arad. While it's a bit of a desolate junction, it's only 2km to walk back into into Sde Boker village if you need a shop.