I've got a couple ideas. Most things currently need to be rebalanced, but I won't address that here.
Ship differentiation. Right now there is really only one choice to make when you pick ships. Do you want a lot of guns, or do you want more fighters? Shields an hull differ a bit between ships, but it doesn't really drive the choice. I think the best way to address this is with the speed stat. Make the smaller ships faster, then also add a module to increase speed and maneuverability. This module NEEDS to be a % increase per level so that you can't just slap a few on the larger ships and have them be really fast. It should also be carefully balanced with missile speed so that only the smallest ships can actually outrun missiles. This is great because smaller ships will be able to dodge some attacks. That means that weapon accuracy is a more important stat. Now you suddenly have a bunch of ways to make a ship worth using. Want a fast ship with a bunch of guns? Fine but it will be fragile. Huge Carrier with a ton of health and point defense, might be slow and lack main guns. A ship with main guns AND fighter slots, might lack point defense or health. If you wanted this last ship, it would do a ton of damage, but you would have to protect it with tankier ships. This is fantastic because it encourages the player to mix and match ships to find the combinations that best suit their playstyles. The way things are now you just choose the ship you want and use three. Related to this, why can so few ships be called in as reinforcements for the fleet? If the player wants to, they should be able to save up for a second armada.
I think the fleet point system discussed earlier complements this excellently. If the player wanted to maybe they would decide to use a fleet of 7-8 smaller fast ships instead of 2-3 capitals.
I also have an idea for a mission type. It would be really cool to participate in a larger battle. I'm picturing jumping into a battle that is already underway. Allied ships and enemy ships are tearing each other up and the player has to decide how best to tip the battle in their favor. The battles should be large enough that if the enemies start to win things can get really out of control. For example: You get the mission to assist allied forces in attacking a sector. When you get close you realize the battle has already started (this could be based on the number of turns it takes to get there or something). On the starmap, you can see enemy and allied ship positions, but not how well or how poorly the fight is going. Let's say you decide to jump in right next to 3 enemies and 3 allies, when you land in the sector it could be that all three allied ships are nearly destroyed and you are about to have a tough fight. It could also be that your allies are doing well and you dispatch the first few enemy ships quickly, gaining you a strong foothold for the rest of the battle. This would be amazing with your "distant battle" backdrop.
This kind of engagement opens up all kinds of interesting gameplay. Maybe you bring a bunch of repair drones and use them to repair the allied fleet. Maybe you position yourself between two ships already exchanging fire to soak up damage for the allied ship. Positioning and strategy become critical in an engagement like this. Maybe make the mission reward dependent on how many allied ships survive to further encourage the player to work with the allied ships.
I think another player mentioned previously how important repair drone behavior is. I agree, they need a few AI tweaks so that they prioritize repairing main fleet ships. Could you also please make repair drones use a green beam? This would make it possible to distinguish them from laser drones (which I'm assuming will be rebalanced to be useful), and make the beam color consistent with the repair beam item. While we are discussing this, something should be done to distinguish the repair beam from repair drones, why should the player choose one over the other? It shouldn't be because one is just better than the other (right now the beam is weak). I would suggest making the beam more effective but shorter range than the drones. Again this encourages the fleet commander to think about positioning.
A final suggestion. I like the way the game is split between realtime combat and rts at the starmap level. It does feel a bit strange though that there isn't really a way to lose at the starmap level. There is no sense of urgency. I want to have to worry about my stations being destroyed if I don't protect them. Enemy ships should try to cut you off if you overextend. Make the commander sweat!