What Lee posted is exactly what I had to do in exactly the same situation you encounter. The last damn hinge screw in an Arts & Crafts end table I built snapped. And I even "pre-tapped" the screw hole with a steel screw and the POC still broke. Why do hinge makers included such inferior screws! But I digress...

Get one of these extractors and IIRC they're run in reverse on your drill. I then drilled the resulting hole a bit larger to make way for a white oak plug. The good news is that it's covered by the hinge. Bad news it just takes longer to finish things.