I'd consider just going with 3D noise and evaluating it on the surface of the sphere.
For gradient noise which is naturally in the domain of the surface of the sphere, you need a regular pattern of sample points on the surface that have natural connectivity information, with roughly equal area in each cell, so you can interpolate or sum adjacent values. I wonder if something like a Fibonacci grid might work:
I haven't chewed through the math to determine how much work it would be to figure out the indices of and distance to your four neighbors (I don't even know if you end up having four well-defined neighbors in all cases), and I suspect it may be less efficient than simply using 3D noise.
Edit: Someone else has chewed through the math! See this new paper on Spherical Fibonacci Mapping. It seems that it would be straightforward to adapt it to sphere noise.
If you are rendering a sphere, not just evaluating noise on the surface of a sphere, and are fine with tessellating your sphere to the resolution of your noise lattice, you can create a geodesic grid on the surface of the sphere (a subdivided icosahedron, usually):
Each vertex of the sphere can have a randomly generated gradient for gradient noise. To get this information to the pixel shader (unless you want straightforward interpolation like value noise), you may need a technique like this article's wireframe rendering with barycentric coordinates: do unindexed rendering, with each vertex containing the barycentric coordinates of that vertex in the triangle. You can then read from
SV_PrimitiveID (or the OpenGL equivalent) in the pixel shader, read the three noise gradients from the vertices based on what triangle you are on, and use whatever noise calculation you like using the interpolated barycentric coordinates.
I think the most difficult part of this method is coming up with a scheme to map your triangle ID to three samples in order to look up the noise values at each vertex.
If you need multiple octaves of noise or noise at a finer resolution than your sphere model, you may be able to do a coarse geodesic grid with vertices and do a few levels of subdivision in the pixel shader. i.e. from the barycentric coordinates, figure out which subdivided triangle you would be in if the mesh was further tessellated, and then figure out what the primitive ID and barycentric coordinates would be for that triangle.