I measured the ball on the suction cup base. It is 16mm or 5/8".
McMaster-Carr sells a variety of 5/8" plastic balls, but they come in packs
of 25 or more and I didn't have much use for the extra 23-24 pieces, I
looked locally.

I was able to find 5/8" wood balls at Hobby-Lobby and Michaels for a couple
of dollars They fit well into the socket on the back of the Oudie cradle.
They are an alternative to modifying the cradle that comes with the Oudie,
allowing a ball to be put on the end of whatever arm you currently use. The
pieces at Hobby lobby have the ball on the end of a 5/8" cylinder making
them particularly easy to mount. Painted black, no-one will know they are
wood.

John