This should ideally be done by a violin maker/repairer (Luthier). However if the instrument is not too valuable, you could try a wire coat-hanger. Bend the end with pliers to make it blunt and very carefully insert it in the LEFT f-hole. You may be able to push the remains of the button out. Don't use excessive force and take care not to damage the instrument. If it works replace it with an EBONY button of a correct size. It should be a fairly tight fit and go in all the way. Don't use any glue of any kind.
Jim, January 2011
edit
I can answer this question...
How do I replace the button at the end of the violin?