My new bike helmet mirror mounts required poking a 3.6 mm hole through a 10 mm polypropylene ball:
Although how I did it worked, it wasn’t pretty.
I had a Micromark Spherical Object Drilling and Finishing Vise which was obviously intended for smaller holes in less challenging objects:
Given the angle between the two plates, I didn’t see any way to put a large hole though the center of the ball:
A scrap of wood aligned the two plates somewhat better:
With that as a hint, the Box o’ Brass Cutoffs disgorged a better spacer, although the original screw was just an itsy too short:
Grabbing the modified vise in a machinist’s vise got me most of the way toward the goal:
Polypropylene is grabby, so the drill stuck / rotated the ball inside the vise / made a mess:
A close look at the top picture shows the nasty ring around the hole (on the right side). The vise grips the ball between two holes punched in the metal plates, contacting it only at the right-angle (-ish) edges forming two rings, so there’s really not enough friction against the plastic to hold the ball in position and any slippage results in a gouge. Perhaps pearls / beads / jewelry behave differently?
Fortunately, I had a bag of 100 balls, so a few failures gave me enough of a clue to do what I should have done from the beginning:
That’s silicone tape wrapped around a ball grabbed in the lathe chuck, with a center drill in the tailstock. There’s barely enough traction between the ball and the chuck to get the job done, but it worked out well enough to build a few new mirrors:
There’s obviously a better way, although it took a few weeks to shake out the solid model …
The Smell of Molten Projects in the Morning 
Having just replaced one, that ball on a stick looks similar to the lever arm on a bathroom sink drain assembly. Your solution looks a bunch lighter and a much more interesting journey to make
Now that you mention it, I wonder if drain balls are any standard size?
Of course, I could just buy a replacement lever + ball the next time I need one … [grin]
I made a couple of mounts a few years ago the same way from ball bearings. I annealed them first and then drilled and tapped on the lathe. I had hoped it would work and fortunately it did.