Here is what we are dealing with; Old brittle plastic strap buttons broken off in their holes.
The trick to pulling out the broken off strap pin is to drill a pilot hole to scres a screw in that we can grab onto and pry / pull out against a "see-saw" type fulcrum.
Here 's the screw I use. It has a machine thread, nut and washer on one end and wood screw threads on the other end. When I screw this screw into the pilot hole I will leave enough room under the washer to get my pliers onto, and then using a piece of a clothespin as a "see-saw" fulcrum I will rock the pliers on the fulcrum and the broken pin pulls right out. We must drill a pilot hole big enough for the screw threads to cut / thread down into firmly. If the pilot hole is too small then the screw shaft and threads will spread the broken pin and push so much pressure upon the hole that the broken pin will either be frozen in place or will break up which would make it neccessary to dig the rest out which is what we are trying to avoid; We want a nice clean pull out:
The nut and washer will run down their threads and bottom out providing a locked nut by which we turn the nut and screw down into the pilot hole until the washer stops flush with the pin's hole. Further turning down of the nut turns the screw down into the pin pulling the pin upward and loosened until it's against the washer. Sometimes the pin can be then pulled out by hand or by simply gripping the nut with pliers or vice-grip pliers. If the pin is still tight in the hole when it's up against the washer, then we proceed as shown further below.
I turn the nut back out until there's enough room to get needle nose pliers under the washer while resting on the clothespin as shown below.
Wiggle the pin a little bit and it will start to lift even if it has a strong grip on the hole.
Applying further wiggling leverage will pull the pin right out. It's also rather easy to envision other similar methods with different kinds of screws to grip onto and pry upward:
