The one you see most often from modelling tools is made by starting with an icosahedron and subdividing each triangle. As I think you're getting at, triangles aren't very good cells for a strategy game.

But, if you take your subdivided icosahedron and truncate it (http://en.wikipedia.org/wiki/Truncated_icosahedron), then you get an awesome shape sometimes known as a buckyball, which is made up of lots and lots (number depends on how subdivided the icosahedron was) of hexagons (which are great for strategy games) and 12 pentagons (which are rather inconvenient). However, if you can live with those pesky pentagons somehow, then this is a great option.

If you want square cells (or just a simple life), then as AllEightUp says, take a cube, subdivide it, then warp it into shape. When you warp your cube into a sphere, the squares become misshaped and inconsistently sized. This approach http://mathproofs.blogspot.co.uk/2005/07/mapping-cube-to-sphere.html is slighty better than the obvious approach of just normalising each vertex.

I second the buckyball idea. Actually, you could use triangle tiles to draw it (just squishing the tiles that make up a pentagon a tiny bit), but for game logic purposes use the center of the pentagons/hexagons as tiles. Or alternatively, if you want all your faces to be the same size and shape (but irregular triangles) draw a pentakis dodecahedron and use the vertices as game positions. Either way, each of the 32 "game tiles" still has 5 or 6 neighbors.

Look at the sphere on the extreme right in this: http://sol.gfxile.net/sphere/

Take two triangles, they form a distorted square (where triangle join at the diagonal of the square).

Perhaps you can distort your square tiles in the same way and use them to tile the sphere?

If you don't care about non-uniform tile sizes/morphology you can always try a quad sphere.