4. Character/ship creation
This was never an option in previous games, but I believe that character creation and ship design possibilities will add greatly to the roleplaying side of the game. People want their character and ships to be unique. When in communication with other pilots in an MMO universe, you want to see a recognizable face at the other end of that communication link. And when your ship is blown to smithereens by another gamer, you want to be able to recognize his ship later on when you have acquired a new ship and are ready to wreak vengeance upon his sorry ass! If letting us design our ships from ground-up is too much to ask, then at least let us customise them extensively.
5. Bring back the classic ships
And while we want customisability, we also want the universe to feel familiar to us. We'll certainly have Cobra Mk III, but we also want to see other classics, such as Asp Explorer, Gyr, Wolf Mk II and others that sparked our curiosity and imagination when we first experienced the Elite universe. Certainly it will be great to have new and improved ships, but the classics are the ones that will make the universe feel familiar to us old hands. Certainly some of the design and detail will need to be modernised, but the ships should still be recognizable.
6. Crew mechanics/management
Larger ships require larger crews to fly. Be it a co-pilot or an entire engineering crew, but they are still additional people on your ship. Let them be more than numbers or skill sets and allow some of them to have their own stories to share. Let them man our laser cannons while we fly the ship. Let us build their loyalty or turn them disloyal by how we treat them. Let us mourn their passing when they die after being infected by alien bacteria thousands of light years from the nearest civilized system. And, for goodness sake, make them question our motives when we hire an all-female crew based on their looks rather than their skills and abilities (editor: this last particular suggestion is not seconded, oh no, not at all).
7. Living world
Let there be local in-system politics leading to civil wars and let players take part in them, running missions for either side, try to play the diplomat to bring peace to the region or cause more and more havoc to escalate the trouble and bring neighbouring systems into it, or force the Galactic Navy to send some ships to the region to bring back peace, and possibly forcing the player to move out or be arrested, destroyed or assassinated. If and when we gather fame as peaceful diplomats or ruthless terrorists, let that reputation keep with us and affect the way the game responds to us.
CONFIRMED ON 30th NOV 2012:
8. Monty Python-esque humour
The original Elite had humour in it that was lost in the sequels. There was, for example, a character called Monty, a Python captain, and many other references to Monty Python, Douglas Adams and their brand of humour. You could find this in the manual and in the descriptions of many star systems and aliens inhabiting them. Let Elite: Dangerous not take itself too seriously and let us have some humour to enjoy while we explore the galaxy and admire its wonders. Let us meet this Monty, a Python captain!
Thargoids and other alien species were pretty much non-existent in the sequels, while Elite itself had hardly any planets with only humans inhabiting them. The battles with Thargoids were some of the most interesting in the game and their single-minded determination to keep attacking all the other races was mysterious enough to provide many chances for interesting stories. But don't let the confrontation with Thargoids (or any other alien species) be too easy. Keep their mystery, even if you do let us see a glimpse of their natures and motivations as we gather evidence from their existence and discarded home worlds, destroyed ships and pieces of technology.