If it's really important for the points to be evenly distributed, and you don't mind doing a lot of calculation to get them that way, you can start with a randomly distributed set and then iterate over the entire set repeatedly, allowing each point in turn to make whatever small adjustment improves your chosen definition of uniformity, and repeat this until the set of points converges. If you're even pickier than that, and not satisfied by just a locally optimal arrangement, the canonical next thing to try is simulated annealing.
For picking points at random, I agree with Peter Shor that taking the time to implement a one-to-one volume-preserving map from a product of intervals to a high-dimensional sphere would be much more wasteful (of time; you would learn a lot) than throwing away 98% of your random numbers. It's an interesting question, though, whether systematically chosen points in a product of intervals can be well-distributed under one of these volume-preserving (but distance-destroying) maps. The first interesting case of such a map is the axial projection from the curved surface of a cylinder of height 2 and radius 1 to the surface of the unit sphere it contains: projecting straight to the axis, one direction gets stretched out in exact counterbalance to the compression of the other direction. Call the coordinates of the cylinder surface z ∈ [$-1$, $1$] and θ ∈ [$0$, $2\pi$]. Choosing an ordinary regular rectangular grid in z and θ does terrible things to the projection. On the other hand, for any $N$, setting zi = $(-N+2 i - 1$)/($2N$) 1)$/$N$ and θi = $2\pi (\phi i$ mod $1$), where $\phi$ is the golden mean, actually gives a very nice distribution of points after projection. It's possible that in any dimension there is such a lattice in the cube that projects nicely, for any N, to the sphere.