Your descriptions leaning on chaos theory and quantum mechanics gave me a few ideas.

First, deterministic chaos means that slightly different initial conditions produce different results -- so why not use the initial conditions as the designator for each universe? I'd start looking at ways to encode coordinates. Something like (x,y,z) is going to produce cumbersome names and you'd have to go out to a ridiculous number of significant digits, but there are other conventions like representing an RGB triple in hex or the astronomy RA,Dec. Some kind of riff on them could be neat, and you could abbreviate the coordinates when there are only a few universes under discussion. But it almost sounds like a more important item is some kind of distance metric for the degree of separation between universes -- maybe where it's <<1 they are close to combining, and when they are >10 or >100 they are extremely well-separated. For that metric, I'd come up with a compound unit that is distance-y but add in something energy-related for the quantum aspect. (As an example, in my con- universe the rating for warp drive capability comes in TqV -- teraquark-volts, tera- to imply large and quark-volts in analogy to the unit of energy electron-volts.)

For specific universe names, I like that you brought up the term "strange attractor." What about coming up with various two-word descriptors inspired or drawn from chaos theory? This universe is the Pendulum Attractor, this one is the Quantum Bifurcation, etc.? For the Core worlds, based on what you've described of them, they are obviously going to pick words like "core," "prime," and so on. For the name of our world, I guess the first question is, whose culture names it?