I think the name "ball arithmetic" is Joris's idea. I don't have a good reference at hand, but I believe the idea of using a centered form of intervals is as old as interval arithmetic itself. Ball arithmetic has been used quite a bit for complex interval arithmetic, where it sometimes offer better enclosures than rectangular intervals, and another nice point of ball arithmetic is that it generalizes to more general normed spaces. However, before Joris's and my own work, I think very few people realized how useful ball arithmetic is specifically for arbitrary-precision arithmetic.