Find some 18 Gage solid copper wire ("Bell" wire is perfect) and hammer the end of it flat, about 1/4 inch long or so. Stick the flattened end in the hole, insert the screw into the hole and turn slowly until the screw threads start to engage. If they won't catch at first, you may have to flatten the wire a bit more. Once the screw does engage, don't reef it down like you did the last time . . . When the screw is tight, flex the wire to break it off.
As an alternative, with an "extra pair of hands" around, have the other person hold the round end of a small ball pein hammer against the back side of the hole - this will act as an anvil. Use a flat-nose punch (centered on the hole) and tap lightly with a larger hammer. This will slightly flatten the metal around the hole, causing it to "shrink". Again, use care re-installing the screw.