Replacing a broken strap button

It has a tapered shaft.  If it's broken off your only option is to drill it out, but it would have to be drilled for the proper diameter for a new one.  I've never done that, I'd bring it to someone who has. 

Typically the hole goes thru the end block.  Theoretically, you could knock it out from the inside with a 1/4" rod, although I have no idea what length or how you'd get enough force on it to knock out the endpin.  Just a thought.