Personally, I would like to see a show based in the main timeline, but something like 100+ years after the destruction of Romulus, and the following war between the Klingons and Federation.
I don't know exactly what sort of premise, exactly, but here are some of my ideas:
1. The Federation and what's left of the Romulans are finally at peace. Vulcan in particular has extended its helping hand to surviving Romulans and helped them found "New Romulus" Some Romulans accept the ways logic like their Vulcan cousins, others remain true to their Romulan ways, but less aggressive and in fact feel (if grudgingly) a debt of honor owed to the Federation.
2. The Klingon Empire and the Federation are now at an uneasy peace, but not allies. The Klingons lost the war (as fought in Star Trek Online) with the Federation, at a great cost to both sides. Efforts exist on both sides in re-solidifying the alliance the Klingons and Federation once shared, but there are elements on both sides that oppose such actions (some Klingon houses bitter over the loss of the war, and some within Starfleet don't trust the Klingons).
3. Following the events of Voyager Endgame, the virus Future Admiral Janeway introduced has decimated the Borg. The Borg Queen suffered her final and true death. This resulted in the Borg losing its sense order and direction, without her leadership. This resulted in the Borg civilization collapsing, much as it did on a smaller scale as seen in Descent. The Borg have splintered into 3 types of factions:
a. The Borg Remnant:
Some Borg are attempting to operate as it did once before, but at a far lesser level of efficiency, and are left rudderless and without direction. They are nowhere near the menace the Borg once were, and often operate in a malfunctioning state, and the fact that the Federation has developed weapons that can now easily repel and defeat them. Rather than conquer and assimilate entire worlds, these have devolved more into raiders of smaller outposts and smaller settlements unable to defend themselves, or prey on wayward vessels that cross their path. They are more akin to marauders, more interested in acquiring raw materials to sustain their dilapidated tattered Cubes and Spheres, only resorting to assimilation if there is a clear and logical reason.
b. The Maxima Culpa:
Some have rejected their machine side and attempted to live more as organics desiring to coexist peacefully with the Federation and other Alpha Quadrant powers. Since these former Borg are made up of so many different species from across the galaxy, and the fact that many of their home planets and respective species have not survived assimilation (and to those that have survived, the former Borg are pariahs), these former Borg have banded together and formed their own civilization, as all others reject them, calling themselves the Maxima Culpa. In fact, these former Borg, the Maxima Culpa, feel a deep sense of guilt and shame of their time as the Borg, and wish to make ammends for their past crimes. They are attempting to reach out to the Federation and countless other worlds that have been affected by the Borg in an effort to assist them using their enhanced abilities and technology to help in any way possible, much as 7 of 9 once did. Yet many across the galaxy wonder if these overtures are sincere, or if this is an attempt at reconquest, so many civilizations, including the Federation are wary of the Maxima Culpa.
c. The Hive:
The third and most powerful of factions of the Borg have "purified" themselves of their organic elements is the Hive. The Hive see organic elements of the Borg as imperfect and unstable, and even see the organic side of Queen having been destroyed by Janeway's infection as the cause of the Borg's downfall. So this new faction has become completely machine, with a new sense of purpose: no longer interested in assimilation, the Hive intends to exterminate all organic life in the galaxy and replace all organic life with living machines of perfection.