In fact TNG and all that followed should have been almost unrecognizable from TOS. Consider the world in 1926 (ninety years ago) and today, that'll give you some notion of how far Starfleet and the Federation should have changed. Transwarp, probability weapons, ships that changed shape, sentient computers handling operations, artificial wormholes, holocolonies, millions of Song-type androids as part of society--that is the sort of thing we should have seen. But that isn't what Star Trek has been about. The template has only ever been fiddled with for the sake of a specific story, then the details forgotten about, the implied consequences (at most) hand-waved out of consideration.
Which is okay. Fictional worlds are rarely if ever that realistic, and dramatically one needs to maintain a certain level of continuity with the audience. Changes are possible, but not many ground-breaking ones, if any. For TNG we got holodecks and the notion of "modulating" shields. That was pretty much it. If you paid close attention you might notice transporters have a greater range.
For a new Trek I suspect maybe one big technological change might work. Perhaps Starfleet ships a few decades after the Dominion War would have cloaking devices--and of course everyone's sensors would have improved so it isn't quite that much of a game changer. Or perhaps the ships will have sentient hard drives, a la Blake's 7, Andromeda or Farscape, so the ship is viewed as a member of the crew. But the more radical changes you make to the background's template the more you run the risk of alienating your fanbase. Mind you, maybe that is the sort of thing you need in order to revitalize the franchise. Nor need it be technological. When Doctor Who returned, the idea of the Doctor being the last of the Time Lords proved a powerful story element that really did change the nature of the stories. So too was the focus on the Companion as a co-lead of the series. The re-imagined Battlestar Galactica made little or no attempt to copy the original, but that was a complete re-boot rather than a continuation.
So--just to stay on topic--here is an idea along those lines:
STAR TREK: YOUNG BLOODS
Fifty years after the Star Trek: Nemesis, the Alpha Quadrant is rebuilding. After a hideously expensive conflict with the Borg, lasting over a decade, entire worlds and species have been wiped out. Others are refugees, or exist only as colonists. On the plus side, the Major Powers have generally good relations with one another. But a certain amount of territory was ceded to the Borg, who evolved into a more patient and subtle set of policies. There are even those who voluntarily join the Collective!
The USS Griffin, a light cruiser (analogous to Saratoga or Nebula classes) explores the frontier near Borg Territory. Aboard are three very talented young officers who "click" together much as did Kirk/Bones/Spock (or for that matter Harry/Ron/Hermione and Buffy/Xander/Willow): Lt. Rowena Styles, whose family has served Starfleet for generations, but she is among the last. She sees herself as a future Admiral, but while brilliant has trouble seeing outside the box. Very traumatized in childhood during the Borg War, but has suppressed it. For now. Lt. T'Lev, a rare Andorian/Vulcan hybrid, who views Starfleet and even the Federation with a grain of salt, extremely well-travelled (her parents were diplomats) and one of the best pilots alive. Sardonic, with a strong sense of empathy that makes her rather lonely, oddly enough. Lt. Geoff Kanter, who was demoted then won promotion again. Bit of a rogue, not above lying at the drop of a hat, in a mission puts his own safety dead last--were he not so able he'd be dead. He is the only crewmen the other two have ever met before (although they don't know each other at first). Friendly, flirtatious, and sometimes melancholy. Finding out why is a major task.
The series follows these three on their assignment to the Griffin, a team in effect learning how to work together and garnering a reputation in the process (rather like Buffy or Space: Above & Beyond) while shifting dynamics between the three chart a course to what they will become. In the end, one or more might win their own command, and another might be someone's first officer. Two might fall in love. One might die, or leave Starfleet. Then again, Starfleet might recognize here is something special and all three get transferred to a new ship.
As far as story arcs go, I imagine USS Griffin makes first contact with a race whose remote ancestors controlled a powerful empire of weird technology, but while they themselves know how to operate the tech none of them have any idea how it works. Remnants of their empire are dotted all over the place, much of it outside their own territory, all of it mysterious (as well as many thousands of years old)--and as such of interest to many parties, including the Borg.
Or other mysteries.
