EWWWW! Thats a nasty problem!
I have actually dealt with this particular issue before, and it wasn't that bad to fix.
Got the buttplate off I presume? Probably won't be able to get the old screw out but can try drill it and use a small easy out to remove the remaining screw. Thats an iffy option, but worth trying if you don't like my other solution.
I drilled a hole immediatly beside the broken screw, and the bit was eating treads off the screw itself as I drilled. When I had drilled the wood gone from alongside the broken screw, I pryed the screw reminants into the new hole and removed it.
Then I made an oblong walnut plug, smeared epoxy in the hole and on the wood plug and drove it into the big oblong hole until it was bottomed out.
I then sawed off the protruding part of the plug, fitted the buttplate, and redrilled the hole for the new screw.
Not that bad a job. And it's all under the buttplate out of sight. Good as new!