That looks exactly like what I have, except maybe a different size.

As a longtime woodworker and luthier, this is the first-choice solution. Toothpicks, shaved if necessary are excellent for this. Popsicle sticks too – they're both reasonably hard wood.

If one had the tools I have, then this is what I would do, if for some reason the "hole-tightening" trick above didn't work. Note that threaded inserts (the thing folks were trying to describe above... ) would indeed be a solution, but be aware that getting threaded inserts exactly where you want them can be a real challenge, as is getting them installed in the kind of hard wood that Sticks are made of. They can, and will break as you're trying to get them screwed into the wood. Don't ask me how I know this...

I would go the threaded-insert route only as an absolute last resort. In the past when using threaded inserts for a bolt-on Fender-style guitar neck I'll install the inserts first, then use them to mark the holes that need to align with them. Drilling the aligning holes then installing the threaded inserts to match...well, that way lies madness!

Using a drill press or similar, drill the hole slightly oversized. If the old screws were 5mm, then check a chart for the appropriate pilot hole for threading a 5.5mm or 6mm screw. If the hole is pretty chewed up, you might need to go with 6mm. The chart should indicate various thread depths for different materials. For soft materials like brass or plastic, 75% is recommended, and I'd use that for wood. For that, the pilot hole for the 5.5mm screw would be 4.6mm, for a 6mm screw, 5 or 5.5mm (depending on the thread pitch. I'd recommend using a coarse pitch of 1mm, though a .75mm pitch wouldn't be a problem on these hard woods.) The thing to check after re-drilling the hole oversize that all of the old threads are removed, leaving a clean bore. If the hole doesn't "clean up," then go to the next larger screw diameter. Use a "bottoming" tap for blind holes for the screw size you're selecting. "Normal" taps have a long taper to them, and will often hit the bottom of the hole before the threads get properly cut. A bottoming tap has a small chamfer at the tip, so they're a little harder to get started, but will cut threads for almost the full length of a blind hole.

A quick aside on tap drill bits. In the US tap drills are almost always "number" drills. These drills are in sizes in much smaller increments than normal fractional drills, and it's critically important to use the right bit to give the tap the correct starting diameter in which to cut threads. That 4.6mm bit I mention above is probably similarly a "specialty" item in the rest of the world. Those types of bits are usually found only in the better-stocked hardware and home stores, especially as you're not interested in buying a complete set, just one or two bits for a specific job.

After re-drilling and tapping the hole(s), run in the new screws to verify the size is right. Remove the screws and wick a little CA glue (Super Glue) into the threads and let set up. Then take the tap and "chase" the threads which will probably have become swollen by the CA glue, put a tiny dab of paste wax on the screws and run them in again and you should be good to go.

IIRC, the holes in the belt hook are a bit oversize and you might get away with going from 5mm to 6mm without issue, but running a 6mm drill bit through the plastic hook to enlarge them slightly if necessary to accommodate the larger screws shouldn't present any problem.

I mentioned starting out that you really need a drill press or one of those drill-press style attachments to make sure that you don't allow the bit to wander, or be run off-axis or it will "wallow out" the hole possible getting you back to the place you started. Not that I haven't done such a thing by hand, but ya gotta be real careful.

Hope this helps somebody sometime...

Toothpicks. Great idea! I didn't think of that.

Minister? I'm flattered, Gene-O

Thanks for the very full write up SteveS.

I hope I never need to make proper use of it.

I hear ya...