Oh I don't know about specific threads.
I just imagine you'll want them not to strip easily, which suggests a larger thread/pitch. And also you'll probably want enough room such that you have three full thread wraps engaged before you you contact the spherical surface and begin applying pressure, which depending on how much room you allow in your design may suggest a smaller/finer thread.
I'll suggest you design in more room, if possible, rather than accepting to smaller/finer thread. I'm just guessing here, but it seems to me like such things I've seen (I own a couple phone mounts of this type) made out of the appropriate material would probably have adequate strength for hand tightening with 20 threads per inch or less and three engaged before the tightening begins.
Honestly, if you aren't already experienced in this area (thread suitability in whatever material you're working with), this seems like the perfect opportunity to 3D print a prototype of your first attempt and test it. Then make modifications to the design if you discover any weakness.
