What does it mean for a set to be SO(3) paradoxical? I am trying to understand the Hausdorff Paradox for later use in the Banach-Tarski paradox. The Hausdorff Paradox states that: "There is a countable subset: $D$ of $S^2$ such that $S^{2}$ \ $D$ is SO(3)-paradoxical". I am aware of what the group SO(3) is, but am unsure what SO(3)-paradoxical means. Any help would be greatly appreciated.

This is a specific case of $G$-paradoxical, for a group $G$. The basic idea is that a set $Y$ is $G$ paradoxical if it can be partitioned into two distinct subsets such that the action of $G$ on the subsets gives something that looks like the original set back. This is the core concept underlying both paradoxes that you named.

You can read the formal definition at the bottom of the first page here.