For whatever it is worth... 

- A new ship design. We're probably talking about something not too far from the basic layout as seen in TOS, VOY, TNG but I'm hoping for some cool twist. Of course, odds remain Star Trek Online has already done nearly anything, but still...
- Ambiguity. By that I mean more than simply "no easy answers" but also an exploration of tensions between equally valid points of view. I would argue for example "In the Pale Moonlight" (DS9 S06E19) one of the best episodes ever, because it explored a terrible moral cost for the sake of a higher good, eschewing formula in favor of a disturbing 'open' ending.
- Genuinely complex characters. I could point to other shows--some of them space operas even--but instead note that Lieutenant Ro Laran of TNG and Garak of DS9 remain among my favorite characters in TREK, and that the original trio of Kirk/Spock/Bones remain a standard for how real characters interact. Way too many characters in later shows depended pretty much entirely on excellent actors (Patrick Stewart, Robert Picardo, etc.) to make up for the lack of good writing.
- Less Technobabble. Notice I said "less" rather than "no." The whole particle-of-the-week thing got very weary very quickly, with increasingly the resolution of plots depending on imaginary science a-la-Batman's-utility-belt rather than decisions by characters.
- Drama. As opposed to melodrama. A very fine author (Stephen R. Donaldson) described once a quick way of distinguishing between the two. Imagine three 'roles' in a story--Victim, Villain and Hero. Obviouisly the Hero rescues the Victim from the Villain in some way applause applause curtain falls. But that remains mere melodrama, because in drama characters change roles. The Victim rescues the Villain from the Hero, then the Villain rescues the Hero from the Villain, and so on.
- Well rounded alien civilizations. Sorry but I've found it all but impossible to believe in several races in TREK. Klingons and Vulcans in particular seem to be just one thing each as a culture, with Ferengi even worse. But that was nothing to the void we got where the supposedly ancient and complex Bajorn culture was supposed to be. The Cardassians at least seemed like a complete society, to the point where we knew what their novels were like, how the approached family, their sense of architecture and how government was supposed to work, etc. Okay, the Vulcans grew to be tantalizing in ENT, which also for the first time gave us a lot of lovely hints about Andorian culture. But I'm hoping the new show breaks away from the route of an alien race with exactly one dominant trait or flavor in favor of ones that feel real. Part of this would mean some genuine exploration of a such a culture.
- Religion. Could we please allow people of faith to exist on their own terms without falling into fanaticism and superstition? Not that religion should end up promoted in some way, but treated with respect rather than stereotypes and condescension? This one I suspect is the one that would cause the most controversy--fans of BSG for example by the last season kept complaining that any members of a high tech culture would take a religion seriously, and that to do so was suicide. Well, no. That is not true. Frankly the sneering attitude towards all religion in TREK gets on my nerves, not least because it avoids all mention of any religion humans might still follow. Not that I'm asking for the fuzzy mystic storylines of Space:1999 or Buck Rogers!