It's funny that the above suggestions include exploring a previously unexplored region of space (Voyager) or featuring all-out war (Deep Space Nine). Neither option proved particularly popular when done before. I'm not trying to open discussion on creative merits, but it's certainly relevant to mention these facts. Next Generation was conceived to feature all-new threats, but other than the Borg, none of them were nearly as well-received as previous ones. No one is ever going to argue that the Ferengi were the second coming of the Klingons. Gene didn't even want Worf in the series, but he ended up helping redefine Klingons in the franchise.
Which is funny, because another popular option has been a Klingon series. It's not new aliens or a new century that fans crave, but storytelling they find universally acceptable. And yet, fans of any extraction are notoriously hard to please. Once they find one gold standard they find it incredibly hard to accept even the idea of another. Imagine the Chicago Bulls trying to say, "We found the next Michael Jordan." Twenty years on and...? Other franchises, other stars, but no one has found the kind of cultural impact as Michael Jordan. It's one of those generational things. Not that I want to get into that again (different thread).
It's also funny (and somewhat relevant to what I was just saying), @SimonJones referencing IDIC in such a casual way. Everyone involved in the original series hated the idea of Gene shoehorning that concept into one of the episodes, because supposedly he was just doing it as a marketing ploy for some pendant he wanted to sell. ("Merchandizing, merchandizing," as the parody of another franchise suggests.) But times change. Maybe one of those ideas the franchise already used can work in a new iteration...
Me, I love the deepening of the lore. I loved the Klingons in Next Generation. I loved the station life of Deep Space Nine. I even loved the Andorians in Enterprise, and yes, the Borg in Voyager. If you get bored learning more about alien cultures, chances are you're not the kind of person I want to hang out with anyway. I love the curiosity factor. Fans like to think curiosity comes from exploring the unknown, but there's a lot of template episodes dealing with the unknown, and there's a reason episodic storytelling has been supplanted in the popular culture, and why fans clamored for so long for Voyager to go full serialized, and why Enterprise's final two seasons, featuring plenty of serialization, were its most popular (among fans, anyway). This doesn't mean a new iteration can't devote itself to one new thing for an entire season or its whole run. That would be pretty interesting. But saying that Star Trek hasn't attempted it already (the Gamma Quadrant, the Delta Quadrant, the Expanse) is just wildly inaccurate.