I'll most likely need to re-run. For the first batch of prototype, what is the safest number? Ordered 10pcs of PCBs and ordered 10x components.
If you are trying to minimize your costs, ten seems a bit excessive. If you only need to get one board working to verify the design, then two or three boards is a reasonable minimum.
You need more than one board so that when you find a problem during debugging you can check whether the same problem exists on other boards. If you don't find the problem on the other boards, then it is likely a manufacturing or component defect on the non-working board. If you find the problem on all boards, then it is a design problem. Statistically, three boards gives you a more definitive answer to the design error vs manufacturing/component error question than two.
I'll most likely need to re-run
No, you will need to respin the board. Get over yourself and learn to be realistic.
Even if your circuit is perfect and you don't find something else you wish was connected to the main controller, or need to use a different pin, or whatever, think of all those morons in sales and marketing that will change their mind once they see a physical thing. Someone will want a different box, a extra button, the LED in a different place, etc.
Has this thing passed emissions testing yet? How about a shake table or environmental hot/cold humid/dry testing? If you still think the first spin of a board that is complex enough to include a 256 pin BGA is ready for production, then you are way too arrogant to be a good engineer.
As for how many first prototypes to make, that depends on how many working boards you need for the first phase of firmware and software development. Each engineer working on firmware probably needs their own board, plus a spare, or two if the team is large enough. If it's just you that will be working with the board, I'd get no more than 3. One to use for development, one as a spare in case you blow up something, and one more so you're not stranded when someone in another part of the company "borrows" one to show off, tries to get support for the project, etc.