I have thoroughly enjoyed playing the beta, warts and all, but I think there are several relatively simple ways to deepen the strategy and tactics involved. Many people have talked about specific capabilities and additions, but I don't think that the current game can be properly judged without playing multiplayer or against a finished AI.
I'll start with my wacky idea. I think someone else has mentioned it, but I'd like to remove jump lanes. I can already see the horrified looks on your faces. Honestly though, I think it would add much to the game. There would be a possibility of deep strikes at an enemy's core, more flexible attack plans, and most importantly a front that is too big to totally protect. You'd need to leave certain systems uncovered to have any offensive operations. For a better defense I could see phase 'exhibitors' (like interdictor cruisers in Star Wars) that pull ships out of phase space, There could be asteroid belts and nebulae that restrict travel. Ships could have jump limits that are increased by research.
A less wacky idea is to give ships the ability to move at sublight speeds between planets. That way, I could launch an undetectable attack on someone, but it wouldn't come to fruition for a long time, and it would leave me vulnerable while my fleet was in transit. You would need to research something like 'Huge Fuel Tanks' or 'Cryo Sleep' beforehand.
Ships should be able to turn off all betraying emissions and disappear from sensors, unless they are extremely close to the enemy. Granted, other races might be able to do this, but I think even the TEC can turn off their ships. This would allow interesting ambushes and surprises, letting an outnumbered person overcome a massive fleet. While ships are 'lying doggo' (quoting Honor Harrington for you Ron Lugge) their shields could be down or something. If ships could lie in wait fleets would have to have scouts out, to detect an ambush. Maybe Arcovas could detect hidden ships from a much farther distance, to make them ideal as fleet scouts.
Counter PSIDAR should be able to be researched. If someone can cPSIDAR level one, it reduces the enemy's vision of their fleets by one jump. So if your cPSIDAR is higher than someone's PSIDAR, they don't see your fleets approaching. Maybe after PSIDAR level two and counter PSIDAR level one, the costs go up exponentially, but those two technologies can be infinitely researched. I'd like to have one technology that resources can be sunk into, but costs and research time would go up so much that it wouldn't really be a race, just a strategic decision. So you'd have to decide whether a larger fleet and with better shields for example, is worth more than the enemy being unaware of your fleets approaching their planets.
If your enemy does get ahead of your in the PSIDAR race, there should be a way to recoup. People have already mentioned the possibility of ship debris, but I'd like some so that salvage cruisers could get technologies off them. For example I salvage an enemy kodiak that has repelon armor and cPSIDAR 2, maybe I get half of each of those technologies making it easier for me to research them. I realize that this won't work if you salvage a ship from a race that has an entirely different research tree, but perhaps a salvaged Vasari ship would give the TEC some weapons research (I don't know how it will work). If salvagers existed, it would create the interesting tension when you have a fleet that is losing, do you stay to destroy your debris to deny your enemy your technology, or do you save as many ships as you can? There also might be a technology that gives ships an auto self destruct to deny salvagers.
Other people have mentioned regular ships getting experience and increased functionality, and I agree. Also GreatEmperor mentioned a ship becoming a captain, and granting an aura to nearby ships, I'd like to see the most veteran ship of any fleet become the captain or flagship. I would like some kind of morale, Rome Total War style if possible, to put an end to ships approaching one another and happily stopping and shooting until one side is gone.
These are most of my ideas on creating more strategic choices without too much unnecessary complexity. Tell me what you think, and any ideas you guys have.