Being from the Charleston area, I know all of the campgrounds in the area. Their actually is a campground at Beckley. However, it is probably 25 minutes off the highway and a pretty tight squeeze to drive to and to get in. It is called Lake Stevens and here is the website.
http://www.lakestephenswv.com/ We have stayed there many times, but I would probably not use this as an overnight stop. But check it out. As said earlier, Pipestem State Park is about 20 minutes south of Beckley and probably 30 minutes off the interstate. I have driven through the campground but we have not stayed there.
http://www.pipestemresort.com/ Your best bet is probably Camp Creek state park. This is about 20 minutes south of Beckley and a short trip, 5 minutes off the Interstate. Here is the website
http://www.campcreekstatepark.com/
Back north on I77 at Sissonville there is a campground called Rippling Waters but is an hour and half north of Beckley, probably just 2 hours into your trip. This is about 15 minutes south of Ripley,WV if that rings a bell to you.
http://www.rvparkreviews.com/regions/West_Virginia/Romance.html
If none of this looks good for you, well, there really are no other options in the area. You do have the Tamarack, which a large rest area with all the amenities including a very nice WV souvenir center/convention center. There is plenty of parking for overnight. Other than that, you want to go on south to Wytheville as someone above stated. I know nothing about the campground there.
I guess I am used to the Turnpike from Charleston to the VA state line. I don't mind the curves or the hills. The rough road is what really gets me blood pressure up. I guess when you are used to pulling a 5th wheel in WV, it really is like vacation when you leave the state and get to flat ground! But we love our state! Go Mountaineers!
Sorry for being so windy, but hopefully this will help the original poster.