I have a clip with a completely black background - actually an unlit space. There is a red exit sign showing in one spot. Camera is locked down and nothing in the vicinity of the sign is moving

Logically all I want to do is copy a small piece of the background and move it over and paste it on top of the red sign.

Just a simple static mask would do the trick I think

You can do it with a bezier mask or cookie cutter and track motion.

Duplicate the clip, place it over the original clip, use the bezier mask or cookie cutter to create a rectangle, then use track motion to move it over to the sign.

Bezier mask will probably work better, as it's more precise, and you can control the feather better. Chances are, the "black" is not even throughout the frame, so taking a rectangle from one part and moving it to another will reveal a slight difference; being able to feather the mask will keep it from being an obvious, pasted-in rectangle.

Thanks much - I'll give it a try.

David's answer is spot on, of course, and that's exactly how I would do it. But if you're results aren't perfect and that masked area is still obvious I would then try taking a still of the frame (best/full), removing the exit sign in gimp/photoshop etc (where you have infinitely better drawing/cloning/repairing etc tools), and replacing the video event on the timeline with the edited still image.