The major difference between the original Star Trek and everything that came after it is that the later crews were all stooped heavily in background material, starting with The Next Generation, even if a few of the biographies were taken from past Gene Roddenberry projects. Besides the material that came prepackaged, the second Star Trek crew was also the first to have adventures that actively added to their stories.
Assuming this is ever attempted, and if any Star Trek after the original ever gets this treatment this crew's the most obvious beneficiary, these are the points that would help make a truly stellar movie (series):
- Jean-Luc Picard - It can sometimes be forgotten, because after the first season it was not really considered relevant anymore, but Picard had a command before the Enterprise-D. It was called the Stargazer and he famously lost the ship in the first recognized encounter with the Ferengi. Not only that, but there was also a whole family of Crushers involved, including one we never met (but more on that later). The younger Picard was constantly pictured as more rebellious than the stolid man we would come to love. That journey alone could carry a whole film. But you could also center the story on the most famous incident of his career, being assimilated by the Borg, which was something First Contact revisited because it was immediately considered the hallmark of the series. You can't go wrong here.
- The Crushers - Beverly and Jack Crusher were lovers in a career that doesn't often see that done successfully. Jack died tragically in an away mission under Picard's command, and there was always the suspicion that Jean-Luc had always harbored a crush on Beverly, a potential romance that left its mark throughout the series. Toss in Wesley the boy wonder, and his whole Traveler saga, and you have a completely different and equally full-bodied story to center a reboot on, in a place fans don't often look.
- Tasha Yar - I'll bet part of the reason Denise Crosby vacated the spacedock so quickly was that she was as aware as anyone could be that Yar had one of the most interesting backstories of the whole series, growing up on a ruthless colony world she barely escaped, which was barely acknowledged while Yar herself was in the series. Her sister shows up later and has the episode that could have been done much better with Yar herself involved. Keep the character in perspective and you have a far bigger winner than fans ever realized.
- Data - From his creation to the shenanigans of his "brother" Lore concerning the Crystalline Entity to joining Starfleet to the fight for his very existence, Data is ripe for storytelling potential, a point the series always liked to keep in mind. So there's plenty there to exploit in a reboot as well.
- Worf - Like Data and Yar and Picard and the Crushers, Worf has a tragic origin story (I literally just realized all the tragedies in this story!), his Klingon parents murdered and later framed for treachery, raised by humans who did the best they could but were never, well, Klingons, and being the only Klingon in Starfleet not so long after Klingons were the blood enemies of the Federation. So, reboot material in spades.
- Riker/Troi/La Forge - This is a whole generation in itself. There's the Riker/Troi romance, always ripe for exploiting, and the fact that La Forge was more or less originally Riker's best friend. Clearly that relationship would put both their careers in better perspective. You may notice that La Forge doesn't actually have his own story, because for all intents and purposes his artificial vision was his tragedy. I'd keep the visor in the reboot. Distinctive images are always handy, especially when they already exist. Plus his learning to become a master engineer is distinct from Scotty's version in either guise, and the later bond with Data is another thing that could use better defining.
- Q/Guinan - Whether together or apart, they're essential to the Next Generation story, one as Picard's opposite number and the other his guiding light (a truly interesting reboot might not even make it clear which is which!).
The challenge would be adequately representing all of them. But then you might not have as many people wondering if this is really Star Trek. Because it's all there already, and all of their journeys reflect the central message of the franchise, the trickiness of the human condition and the challenges of the unknown. Maybe center it around Q once again, but this time maybe avoid the whole Farpoint business...