I recently inherited a transfer student with a particularly problematic double-jointed thumb. It may be the loosest, most unstable thumb I've ever encountered. When I chatted with her about it, she clearly knew it was a problem. But the real problem was that she didn't understand the problem. No student can change something they don't even understand.
I patiently explained to her that in order to play the piano well, one's hand has to be stable. Sometimes joints are so wobbly that they can't be counted on.
The most important thing about stabilizing a double-jointed thumb:
1. Help the student become aware of problem in a non-judgmental way.
It's not their fault that their thumbs are wobbly.
2. Show them a way they can work on it themselves.
Do this in small doses, with lots of rewards.
They are not trying to annoy you, even though it can be annoying when that thumb wobbles like a bobble head.
Here's a video to show you one way to a student can teach himself how to play with a stable thumb.
(This is not a video of the new, super-wobbly student.)