I often roll around the idea of a devising Trek characters for a new series in my head. I always thought Kes and her short lifespan would have made great tension as the 7-year series came to a conclusion. So I have often envisioned a character with a terminal illness who will ultimately die by series end. This person is not one to be pitied; with good writing the audience would often forget that they are ill at all. it will be heroic tragedy at the end of the show that despite all the tech and advancements, their life still cannot be saved. A great reflection of the world we live in here and now.
A similar character, perhaps even one combined with the above, would be a retired soldier left from the wake of the Dominion War. They would suffer from PTSD and would constantly be battling their own demons while still trying to be the "good officer".
I've always considered some of the less explored ship department personnel (stellar cartography, geophysics, etc) could be potential leads. One that I've taken personal interest in is ship's Cultural Historian built loosely on aspects of Guinan and Phlox whose alien perspectives often echoed the human condition and episodes like City on the Edge of Forever and Things Past. Ultimately, the theme of this character would be that although pushing into the future, we must learn from history (which often will be our present)
McCoy and Pulaski being divorced always seemed like an unexplored facet too. A divorced character, probably an angry one, would also be an option.
I have many more that I've tweeked over the years. For me, Trek's potential is endless. I was dismayed when plots were getting lazy during Voyager and Enterprise. I see no reason for that, other than new blood was direly needed. I would love to pitch ideas for a new series. If one ever starts it would be great if they re-implemented the outside script policy again.