If you're a wimp and/or cheap, take out the strap button (unscrew it). Get some toothpicks. Jam a few toothpicks in the button hole, then break off the parts that're sticking out. Carpenter's glue (the yellow stuff) is optional. Screw the button back on.

Another slightly more expensive and involved method is to get some good quality wood filler. I have some "wood epoxy" at home, which I have yet to test run, but on the box it says that it is stronger than wood and can take screws. I actually found a link to the very stuff I have. With this stuff, you mix the two parts of the epoxy together which makes a paste. Then, insert the paste into the hole and let it cure for probably about 24 hrs. Then you're good to go; you can drill a pilot hole for the strap button screw and then re-install.

The third way, which I have actually done on my friend's bass, involves a piece of wood dowelling, carpenter's glue and a drill. First, find a piece of wood dowelling that is only slightly larger than the starp button screw hole. Buy it. Bring it home. Then, drill out the strap button hole with a bit which is the same diameter as the dowel. Cut a piece of the dowelling that is the same depth as the hole you've just drilled out OH MY GOODNESSS WHAT ARE YOU DOING, DRILLING YOUR GUITAR HAVE YOU GONE MAD!?!?! . Next, take a deep breath and calm down. Stick some carpenter's glue into the hole, then insert the dowel. Let it sit for 24 hours. Then, drill a small pilot hole for the strap button screw and reinstall.

There.