The main character would be a plucky young lieutenant, playing off of a Pellew-esque captain. The core cast would be very small, preferably some kind of freudian trio (ego, superego, id). The rest of the command crew, as well as other prominent positions, show up often, but don't have the some focus on characterization as the core cast.
All the main characters ought to have good, personal
reasons for being where they are in the universe. DS9 is a good example of this... all of the main characters feel like they want to be there, or have been forced to be there. There's a lot of character agendas at work. Its a good example to follow and improve upon.
Don't be afraid to change up the cast now and then. A lot of long-running shows do this. Think like JAG, or CSI, or some other mainstream shows.
Speaking of which, police/detective/law procedurals are the rage these days, so why not take after them a bit? TOS took a few bites out of spaghetti westerns after all. Add a heavy analytical aspect to the show, lots of laboratory time, lots of investigations. Really chew into the investigative aspect of Starfleet. Maybe have there be more mystery stories than have been in previous Trek series.
More moral and ethical ambiguity. The best decisions Kirk, Picard, etc. ever made were ones where they sat at their desk afterwards and had to tell themselves "I did make the right choice."
Use classic Star Trek setting and furniture. Warp drive, transporters, Klingons, the works. Don't care about the timeline. Prime or JJ works for me either way. Get in on some of the cyberpunk/posthumanism/new space opera wave. RAMA's linked post has some cool ideas in this regard.
Keep time travel to an absolute minimum, since all it breeds are repeats of Yesterday's Enterprise and Tomorrow is Yesterday.
The ship would be an old-as-hell rustbucket that's fallen behind the times. The crew thus, must rely on their wits more than their phasers. Keep technobabble to a minimum, and where there is technobabble, have someone Watson a way simpler
explanation out of the technobabbler.
Don't get bogged down in minutiae and trivia. The show should have something of a mythology of its' on, to make it accessible to new viewers who might not have seen any
Star Trek before. If something from elsewhere in Trek is introduced, it needs to have a natural and smooth expository entrance of its own within the series. It has to stand totally on its own, again, for accessibility's sake. Like if one were to use the Cardassians, or the aftermath of the Dominion War, or the loss of Vulcan, the conflict needs to be established within the context of this new show.
A huge war premise doesn't appeal to me. However, having the Federation embroiled in a couple of smaller conflicts far from the main characters would feel more realistic, would be more dynamic, and would present nice opportunities for modern allegory.
I'd have each season be a self-contained story arc, with standalone-ish adventures contained within. Bonus points if they sometimes end up being part of the main story. Try and shoot for a wide variety of plots, the audience should be surprised each and every week. Tap modern day scifi novellists and short-format authors for scripts.