Mapping spheres is the great mystery of cartographic projection that has never been solved. The problem is that there is no way to map a flat image onto a sphere without distortion.
However, you can come close.
type one:
In a modeling program make a cube that is highly tesselated on a regular grid on each side. Then make six texture that tile perfectly on a cube, like a skybox, and map them to the 6 cube faces. Then you must find a way to make that tesselated cube into a sphere. it can be done simply in 3DSMax using the spherify modifier; other packages may have similar filters. And viola, near-perfect sphere mapping.
Type 2:
If that dosent suit your needs, you can try using two paraboloid type mappings, one for each half of a sphere.
Imagine a standard planar mapping of a sphere, where only the top third or so is reasonably distoriotn free. The edges and middle are of course stretched. So in your mapping method, simply strech the vertecies on the outside outward, to make a sort of paraboloid or squashed hemisphere.