We can see stars but not planets etc.

The biggest advantage to how its currently done is that it is obvious what you have and haven't explored.  The more detail you put into the fog of war the less differentiation between it and the map and the more potentially confusing it could be for players.  So any changes would need to take that into account.

I agree that this initial 'fog' should be nothing more than empty space with just stars (and perhaps large nebulas) showing. As you explore the hexes can become visible denoting 'explored space'. Any space without hexes can simply be unexplored inside this fog.

As for previously explored space in the fog of war - once you leave an 'observed' area the planets/asteroids/units etc can become faded/translucent and remain static as the last way you observed them. This way you can still see the explored areas but you will not know how it has changed (i.e. a previously available planet will still show translucent as available but could have been colonized since the last time you were there - or a translucent enemy starbase will show on your map but it could have been destroyed since the time you were there). These instances would all be true unless you have an alliance, treaty or spies/intelligence (or even new techs) in which the areas on the map can update occasionally or in real time.

I think another forum member suggested a shadowy overlay to the board. I think if done well this could work. In addition to what I stated above the fog of war could contain a very subtle rippling shadow that overlayed the fog areas. It would have to be extremely subtle though, warping space/light ever so slightly that it still looked like empty space but animated/distorted enough that the player would be certain that they were looking into the fog.

I agree that stars should show up on the fog of war, I do like lecek's idea on what the stars should look like as well. Just having that plain circle is boring. Why not make it look nice?!