I think I've read old magazine ads that showed something like you are describing. Somewhat similarly, a very old Stanzel Raider kit has a full length fabric strip glued along one side of the stab/elev. (That kit, by the way, is amazing in the level of prefabrication and exceptional quality.)
If you go the whole length, you need the hinge to be very flexible, so use a thin material. If I was going to give it a try, I'd start with a strip of polyspan. Because there is so much total hinge length it should be quite strong as long as there is no tendency to propagate a tear. The random fiber orientation should be perfect in that regard. I think Tyvek would be too stiff for a continuous hinge, but works for segments and of course, is nearly indestructible.
I would not use any type of linen or cotton fabric. It absorbs paint and becomes brittle. That also stiffens the hinge. When you flex it to free it up you are actually damaging the fibers as they break loose. It will also rot over a long period, should your masterpiece survive until that day.
If you are looking for other alternative hinge concepts, you should check out a really good sewn hinge using a stab/elev construction designed to accommodate this. They can look pretty good--in some cases barely visible.
Dave