Replacing wires with threads?
Conductive thread is simply thread with wire woven through it, so it won't necessarily be any more robust.

The critical thing for reducing fatigue in wires that are flexed a lot is to maximize the radius of curvature. Instead of bending it directly around the hinge, use a generous loop of wire, so that the maximum curvature is very gentle. You may also consider looking for cable that is specifically rated to withstand a lot of flexing.