How can I prevent a ceiling fan light kit from wobbling?
You'd have to take the light or light kit apart. Typically they are locked into place on the wire shaft &amp; quite wobbly &amp; loose.

But, if this has gotten looser &amp; looser, then you should be able to tighten a nut or 2 on the wire shaft to get everything back to solid.

Be conscious of the loosening in the future by holding the light socket with 1 hand while swapping out the bulb with your other hand. You can also put a drop of any glue on the threads to keep it tight a good long while.