The Next Generation: REBOOT!!!

[Samuel]

Yes, but we would need a reason for the First Federation not to have been able to do it first.

[Simon_Jester]

The original Borg were techno-scavengers, right?

Set it up like this:
Our reimagined Borg are also scavengers, and they're not interested in primitive species with no useful technology to scavenge. They can create fresh drones easily enough; why would they bother to "assimilate" a species that has no useful cultural, biological, or technical distinctiveness? If they need resources they'll strip mine planets, but there are plenty of uninhabited worlds in the galaxy.

For now, the Federation and its immediate neighbors are beneath the Borg's notice, so there's not much chance of the entire Borg species staging a mass migration into their space and swarming them under as they did the First Federation. However, that places the Feds in a serious strategic dilemma: they need to become powerful enough to resist the Borg without passing through an intermediate stage of being strong enough to attract notice without being strong enough to fight back, as the First Federation was.

Alternatively:
The FF-Borg war shook the galaxy. Fesarius-class dreadnoughts dueled with Borg Spheres; planets were laid waste by weapons no one else this side of the Q Continuum can even describe, let alone duplicate. The Borg won, but they're still rebuilding. They're more interested in restoring their fleet and replenishing their numbers than they are in conquering what's left of known space. Can the Federation fend off the remaining Borg for enough time to find a weakness in their seemingly invincible forces?

In this context, what the Federation sees as a devastating threat is the Borg equivalent of sending a few scouts out to probe the frontier of the known universe. There are some historical analogies:

To the Greeks, the Persian attack on Athens stopped at Marathon was a major historical event. To the Persians, it was a minor punitive expedition meant to slap the Athenians down for interfering in a civil war in one of the Empire's border provinces.

To western Europe, the Battle of Tours in 732 was the great event that defined the beginning of Europe's recovery from the Dark Ages, and the end of Muslim expansion in Europe. To the Muslims, it was the defeat of a summer raiding party that just happened to roll across the Pyrennes to take a look around. Everything really interesting was going on a thousand miles farther east; who cared whether or not some random general on the edge of the outermost fringe won or lost against a bunch of irrelevant barbarians?

[Big Orange]

The creatively squandered First Federation from TOS seemed very advanced and practically Culture-like (only without the absurb reaction times), so the Borg fully assimilating them would be bad news, and the Borg Collective would be even more powerful than they were in the OT. However the Borg getting as badly decimated as they were in their fight against Species 8472 is a good idea, Simon_Jester, it doesn't paint the Federation and other Alpha powers into a tight corner that only the Q/Organians/Prophets could bail them out from. The new Borg warships after the assimilation of the First Feds could be Cubes but with overt Fesarius-class features: the Borg Cubes are precise mathematical shapes (cubes/spheres/bucky balls) made out of small, interlocking cubes that can act as independent vessels.