Can a visor that continually flops down be repaired? 

Yes -- there are a couple of ways that a sunvisor is mounted to post that holds it in place; the screw and clamp method and the tension clamp method. With the screw; the problem is either the plastic bushing in the visor that visor pin slides on; is wore out; or the post that attaches from the mounting bracket (that attaches the visor to the roof) to the sunvisor is wallowed out and is freely spinning. If it has a screw, tighten it, if that don't work, put something small in it to help hold it and then tighten it. If it is the tension type, clamp it with a pair of pliers (with the visor pin out of the visor).

More about the clamp type: the clamp type (where you can't see any screws in the visor) sometimes flops because the clamp just ain't clamping anymore; sometimes removal of the mount pin from the visor and closing the clamp with pliers works. (don't use the pliers unless you use a piece of cloth in the jaws; it will rip the headliner material on the visors).

We repair and recover sunvisors for most cars.