Some systems exhibit a butterfly effect (a.k.a. chaos); some don't. The butterfly effect is where (arbitrarily) small changes to the conditions of the system can totally change it's future course. The weather is a good example of this. The change caused by a butterfly flapping it's wing differently will amplify itself until the entire Earth's weather is different from what it would have been. But other systems aren't like that. They're more "stable". For example if you change the position of any individual atom in my computer it won't make any difference to the computations I'm running. Other things are predictable just because we don't give time for any changes to develop. For example ball collisions are predictable, but if we study many ball collisions in a row, like a billiards "trick shot", then hitting the initial ball slightly differently will make a huge difference.

You ask about quantum events. For chaotic systems, deviations caused by quantum events will indeed cause a butterfly effect.

So whether or not the brain is predictable depends on to what extent it's chaotic, and to what extent it's stable. I suspect that it's chaotic, in the sense that a small tweak to it could totally change the way a thought process goes. But over time my brain will be predictable "on average". I'll behave in ways matching my personality. Similarly a butterfly flapping it's wings might change when it rains, but it'll still rain more in Bergen than the Sahara.

I don't think this says much about Omega problems. Quantum butterfly effects will (I suspect) stop Omega exactly simulating my thought process, but I reckon it could still predict my choice with very high confidence just by considering my most likely lines of thought.