I would like to know how to choose $x$ evenly distributed points from within an n-ball. I think a formal way of defining this is that we want to choose $x$ points from within the n-ball such that we maximize the closest distance between any two points. As a result, it seems, all points should be evenly spaced and many should be located at the surface. What is an algorithm to generate such a set?
Suppose the closest distance between two points is $d$. This implies that $x$ spheres of radius $d/2$ centered at your points can fit inside a sphere of radius $1+d/2$ without overlapping. Equivalently, $x$ spheres of radius $d/(2+d)$ fit in a unit sphere, and maximizing $d$ is equivalent to maximizing $d/(2+d)$.
So your problem amounts to finding the densest packing of $x$ spheres in a sphere. There are some (approximate) precomputed solutions for $x\le51$ in three dimensions, but there is probably no closed-form solution in general. I guess this is not an answer to your question of what is an algorithm to generate such sets, but searching for "sphere packing algorithms" may find you some useful references.