I just tested my various trackballs with a vernier caliper and they seem to range from 56.8mm to 57.3mm. I have an old ADB Kensington Turbo Mouse, which was 57.3mm, the Kensington Expert Mouse Pro, which was almost exactly 57mm on the dot, and four balls for another old Apple trackball that's now no longer around, and two were 56.8mm, two were 57.3mm.
2.250" is 57.15mm. I don't know if they were shooting for SAE or metric, but either way, something in the 2.250" or 57mm would probably work great for the standard large ball. An American billiard ball is 2.25" by design and a British billiard ball is 56mm, so I'd bet that either would work in most large ball trackballs, but the American ball is probably a slightly better fit. That ~1mm isn't much, but it could be quite the difference.
I'd think the problem with a large optical trackball with a nonstandard ball (like a semi-polished stone sphere) would be twofold- one, the imperfections in the stone would stick on the three points of contact, and two, the material itself would abrade the three points down. For a rollered trackball, if the nonstandard ball is abrasive it would probably start to wear on the rollers as well, albeit not quite as quickly as the optical, especially depending on what those rollers are made out of. My Expert Mouse and Turbo Mouse both have steel rollers, so they're probably good for more abuse than any with plastic, but I'd think plastic ones would suffer a lot of wear fairly quickly with something like stone. Keep in mind that the ball for a large trackball is light and rigid in addition to being smooth, so the lack of weight and the lack of plasticity should both be good traits for not wearing the rest of the mechanism out.
EDIT: I just re-measured the Expert Mouse Pro's ball in inches, and with two decimals of precision I got 2.25". When I can find my micrometers I'll recheck to see how close to 2.250" it really is.
