I just started watching through The Next Generation again and I think I know what would make STO feel more star trek like without having to add a TON of new code...
Drama on your own ship.
Bottom line. Pretty much every show in TOS and TNG focus on the drama going on within the hull of the Enterprise and not so much with what is around it. Sure there were baddies that got shot up or attacked but most of the missions were about something happening on board that they needed to figure out.
I don't mean that there needs to be some big medical problem where you have to spend three hours hunched over a virtual microscope looking for microbes or anything like that, but just something that is able to utilize the system already in place but have your ship be part of the mission.
Maybe you could take the role of the Engineering BO (since all of your BO's are already modeled) and go on some micro mission while at warp that would give you a small buff for 1 hour when you arrive at your destination. That could take the place of watching your ship slowly cruise through space as you warp to your destination while you confirm multiple "would you like to warp" dialogs.
The game needs to remain approachable so adding a lot of required steps in order to play would just alienate new players, but optional missions inside your own ship would be a nice addon that could utilize much of the system already in place.
This is all assuming of course that they finish the ability for you to walk around your own ship

Any other thoughts (that don't involve completely changing the mood of the game)?
Comments
Frankly, the solar wind PvP is very nearly crew vs the ship imho. That map is a major PITA.
The go somewhere and shoot stuff mission is the easiest and most obvious to write. Cryptic needs to start thinking outside the box a little and come up with some other possible mission types. They did a nice job with the deliver supplies missions but there's no real drama to these. The quiz missions are a little better. Maybe combine the two ... have a quiz mission where you then need to figure out how best to deal with the issue and deliver those supplies.
I like that idea too and it is one that has been discussed in the Feedback Section under Missions and Episodes. Take a look when you get a chance.
http://forums.startrekonline.com/showthread.php?t=100038
This isn't a knock of your post just a note to let you know you are not the only one thinking of this.
Brian
One of the things I have noted (now that I am looking for it) is that they haven't used the weapons on the ship offensively at all so far... 6 episodes in. So far it has either been Q, or everybody on the ship going crazy, or some space phenomenon taking over the ship.
I see a lot of calls by a lot of people to change the game so completely that it would pretty much mean wiping the slate clean. I think we can work with what Cryptic has given us... we just need to give them the positive feed back required to see it happen.
Heh, that is what I get for not searching first
But at least I know I am not going crazy. Everybody is!
Speaking of going crazy, I don't think that is necissarily a new condition, some have more issues with confusion then others.
Without going all religious, I always remember this thought
2 Timothy 1:7
For God has not given us a spirit of fear, but of power and of love and of a sound mind.
Although I know many people have many beliefs on many things.
A favorite vid.
Mad Mad World
http://www.youtube.com/watch?v=qksZZ6kzfgM
interesting i could see this in the game. Bo's getting in fights you have to manage them based on your choices you would have different bo powers for each battle.