Aside from an epic branching storyline the two biggest things I want out of a new space sim would be atmospheric missions and something that would be really customizable.
On the atmospheric side of things I think it would be cool if you could transition between space and atmosphere whenever you want, but I don't know how technically feasible that would be. You would probably still need some sort of transitional load like they did in WC3 and 4. I would also like to see cities and canyons you can fly through so that if you have a ground target you might have to actually fly between skyscrapers to get to it.
On the customizable side of things I'd like to be able to add my own music, like GTA 3. But also I would like a built in mission editor as well as a sort of instant action option that lets you pick your ship, battlefield, and how many enemy fighters you will face and what their experience levels are. But even beyond that I would like to see tools come with the game that would help the average person to easily build and integrate their own ships into the game.
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Now if the game is in the Freelancer\Privateer vein then I'd like to see a little more freedom when buying equipment for you ship. Maybe have something similar to Mechwarrior where you have so much space to use. So you could take a light fighter, strip it down to a basic shell before cramming in a big ass power plant, but your restricted to the space on hand so you might end up with no cargo bay.
Alternately I think it would be cool if you were free to attach anything you salvage to your ship. So you could blow a corvette in half and take what remains, engines-hull components etc., and attach it to your ship, expanding your available weapons hardpoints and equipment space. So you could, eventually, build a huge Frankensteinesqe battleship.
It would also be nice to be able to attach your own markings to your ship. So you could have nose art from WW2 or simply a shark's mouth. To go along with that it would also be cool if you could build up your own criminal organization and recruit other NPC or live players to work for you.
I've got a bunch of ideas that are too numerous to list them all hear but these are the biggest on my wish list.