So what we know so far about the basic game idea is that it's an alternative timeline and the core of the game is to fly around in space to meet funky aliens and upgrade your flagship as you go, like in SC2. With this basic setting in mind, here's some thoughts. Scrap, ridicule, alter or pick the cherries as you see fit.
1. Good cool story
Well I went through this pretty much in the story thread already so I won't delve deep here. Basically it'd be super ultra cool to have a good plot with some excitement and emotional attachments to plot characters and races. Which preferably then get horribly broken as people die gruesomely to flaming plasma and supernovas. Ahh the drama.
2. Memorable music
Goes along the story and atmosphere in general. Get good music. I don't know if I can emphasize this enough. It's probably the best way to create a good atmosphere. Being a proud owner of Sins, I believe you have this department well in control. Man, I really mess up my games sometimes because I don't want to leave a grav-well because I enjoy the combat music too much and won't upgrade around properly. And vice versa. Music, it's paramount in making a game feel good.
3. Large game world
Well, we're flying in a friggin' galaxy right? Modern tech should allow to create a preeetty big place if need be. Of course, it's up to the people working the game together how well it'll be fleshed out but the potential is out there. Ideally I'd like to see a big starmap in 3D that you can rotate and zoom in and out as you see fit. Again I'll use Sins as an example. The basic controls there are pretty good. Just pick that and put it into a bigger scale and we're game. Of course, being able to zoom inside a star system in a galaxy level map is probably not necessary.
4. Dimensions!
Part of the game world, really. I'd really love if in addition to the already-large-quadrant there'd be dimensional travel included and in these dimensions you could also travel between stars. Or whatever you find inside these weird, outlandish.. places.
Quick thoughts; dimensions with themes! How about a place where the SPACE SINGS! The void vibrates a melody that reacts to your ship/fleet. You could even travel by creating certain kinds of themes that'd affect your location. Or you could meet aliens who you'd have to fight using rhythm in your attacks in order to win (their favor).
Or how about a place that's lorded BY EMOTIONS! Anything is possible really, like the emotions affecting your ships armor/shields/engines (feeling angry? armor melting or engines on overdrive! feeling happy? ships start to float or shields get a regen boost! etc.) or more on the adventure side, emotions could direct you to proper places in this odd world. You'd have to focus your crew to get anything done. Meeting aliens in this place could get really interesting. Poke them the wrong way.. Fun ensues!
Dimension with THINGS IN REVERSE! Ok, clearly it'd have to be bordered somehow as to what's reversed or else you'd simply cease to exist the moment you enter this funland. I'm a bit dry on ideas with this one actually but I think more creative people could come up with some crazy ideas.
Dimensions with REFERENCES TO REAL WORLD OR OTHER GAMES! I don't know what would be more exciting than accidentally jumping right into the middle of Doctrinal War.. Ideas are practically limitless here. Just pick anything fun and cool and work with it.
Etc. Dimensional travel allows for some really crazy and fun ideas in my opinion, as they're not restricted by any rules we know. Great for adding variety to the game.
5. 3D space combat
Hopefully a meaningful part of the game. Maybe I'll take an in-depth look later on in the actual space-combat thread but as for now I'll just state that I'd be happy with a combat system that's something like we saw in Homeworld. Although I love the old 2D combat of SC2 I don't quite feel it'd work out as primary combat system in a modern day space game. So, your flagship would work like a mothership that you can move and use like the rest of the to-be fleet. It'd be powerful in its own right but by no means invincible so you'd have to work with the whole of your armada to overcome the more difficult battles. You'd be able to make groups and issue different kind of commands to them like fleet formations and aggressivity levels.
Would be cool to take control of a single ship and fly it personally, too. Of course you could lock a squad of ships to your personally commandeered vessel. Epicness takes on a whole new level. Just imagine the combat where you first give generic orders to everybody and then jump into a vessel and personally direct a squadron of Ilwddash Thovengers to take down the dreaded dreadnought command vessel of the enemy while dodging hot plasma. Oooooh yess.
6. Resources, exploration, scanning
The mineral farming was a cool part in SC2 but IMO it degenerated into a grind in the end. Or maybe that's just me after playing it through too many times. I've this feeling that it was there mostly to keep the player busy with a some kind of task through early game. All in all, the idea is still cool and worth preserving. I'd go with a quick-scan/harvest option as you explore different worlds and an optional mini-game attached to it. Maybe send a squad on the surface that'd scout any hot-spots your scanners reveal. Strike team, X-Com style. Rewards could be extra minerals, lore data disks, vague clues about things of importance, pieces of easter egg bonus artifact thingie or something like that. Little bonuses that won't neither force you to collect them but still feel rewarding.
7. Racial encounters
Well this one is given. You kind of can't have a Star Control game without some good amount of alien interaction and the various thrilling discussions old SC2 fans are used to. The system SC2 used is fine and I'd prefer to see it return pretty much as it is. This whole point is a bit useless as I feel fun alien chats are self-evident.
8. Diplomacy & racial actions
Ok ok I know that we can't cram just about every nutty idea into the game. There was a fun pic in some thread about this remote control with gazillion buttons. Just tossing ideas! Aaanyway, depending on how the actual game will turn out, I'm feeling pretty good about an idea with the races having a sort of diplomacy theme of their own that keeps on developing. I'd be cool if the races keep doing their thing - like expanding rapidly, which in turn would affect the neighboring races probably negatively. Or you might have the local merchant race selling stuff that some honor-code race deems unethical. You could affect these relations to an extent by giving advice, forming alliances etc. This kind of thing would add some nice feeling of life to the game instead of it being a static, still world where things just hang waiting for you to pop by and do your magic.
9. Research
Definitely a healthy amount of wacky alien things to keep the brains on the ship (or starbase, or earth, w/e) busy. If there's mineral harvesting like in the original, harvester upgrades. If a strike team like I suggested, gear upgrades. If we find alien stuff on alien worlds, info on those. With discoveries and time, upgrades to flagship (duh). New mineral type data? Space anomalies? Bio-scans? Maybe even language research? It's a big world out there and we know we know little.
10. Misc little things I'd love
-Journal. A must-have for today's gamers. Helps tremendously in tracking what was said in key discussions, homeworld coordinates, alien artifacts etc etc.
-"Hangout" at flagship. Really neat way to add a bit of chillout and depth. You could pop inside your flagship at bridge (or even several locations if devs like it and have time to fleshen it out) and chat with some crew members. Little discussions that might give advice or just add some humor value.
-Difficulty choice. Simple way to keep those space fights interesting. Adds replayability too if difficulty would affect puzzles etc.
-Optional gameplay elements. Little things like X-Com reboot Second Wave did. Male/female voice for flagship computer. You know, a sexy lady telling you about warp engine status or giving shied/hull reports. Possibility to beat the game with/without timer, a must have for old-school vets. Anything that's randomizable, give the option to do so. Random starting location? Random racial diplomacy? Random events? Random encounters? Random tech-tree? Random dimensions?
-Possibility for retro 2D space combat. At least as a mini-game.
11. Just a few ideas for DLC
Well, I'm not a huge fan of the whole DLC concept but I guess it's part of gaming today. Nothing special here I guess but if (as) there will be DLC content, I'd love to see extra races and side quests/events that'll fleshen out the game world. It's ok to have some loose ends with the basic story IMO, just as long as the original story is rewarding in itself. DLC is a great way to tell more about the background of that enigmatic Ultraloid race or to finally give player access to the fabled Moon Device which you only heard about in legends.
And of course, all the basic stuff. Extra skins, voice packs, upgrades, tech, pocket dimensions, planet types etc. I'd just like to see additional content that makes the game bigger, not only shinier.
TL;DR: So my idea of a new SC is a game which works pretty much like the original. Just add some neat things like research, planet exploration, 3D fleet battles, dimensional travel etc. Roll in some quality graphics, slap it with good music and drop a well written story on top and we're looking at a triple A quality game. Yessir.