Well, you have to start by figuring out how the Persian Empire is going to collapse, because the Mesopotamia was really very loyal to the Achaemenids once Cyrus conquered Babylon, so there's not an obvious PoD for Mesopotamian independence. My bet for the best PoD is to have Darius die and Cambyses live - let Cambyses and Bardiya fight a civil war while revolts elsewhere destroy the empire they're fighting to rule.
Russian's right with the whole lack of an Assyrian nobility thing. Even though it would only have been a hundred years since the fall of the Neo-Assyrian Empire, the Assyrian line and culture had been pretty thoroughly defeated then. The more likely thing to happen is a resurgence of the Chaldean/Babylonian Empire, which would only be a couple decades removed from power and plenty of nobles who could win support as a rebel king.
You could have a new Assyrian state rise up, I'm not trying to say it's impossible, but it'd probably be more like the Chaldeans were to the original Babylonians rather than a continuation of the earlier empires - occupying the same land but not the same people. And I would imagine that that a theoretical third Assyrian Empire would move its administrative functions south to Babylon in the end anyways, since the great cities of Assyria were much less important by this time, and if that happens they'll depend on the Chaldean nobility to help rule anyways. So I think a Neo-Chaldean Empire is probably the most likely thing that's closest to what you're looking for.
The effects of such a state are hard to say. I would assume that things would end up kind of reverting back to how they were pre-Cyrus, with Egypt happily independent again, a Mesopotamian state (either the Chaldeans or new Assyrians), an Iranian state, and an Anatolian state or two. Maybe one of them tries to conquer all the others, maybe they achieve a long-term balance of power. For the Greeks, it probably keeps Greece poorer and more locked into petty polis warfare - Persian wealth empowered Greece as much as any sarissa or dory did, and there's no great unifying tradition of defeating Darius and then Xerxes. Perhaps this makes Sparta, the polis least dependent on wealth, more powerful long-term in Greece?