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What is the best offshore bank for online business?

Welcome aboard :)

Well it depends a little on what payment processor you want to use? I see on another thread you opened that you are looking for a merchant account and payment gateway! If that is what you want it will limit your options a little.
 
I'm doing stuff on elance and odesk mainly taking paypal payments now. I want to build a website with order tracking etc. so that when a customer place an order he creates an account on my website, pick his order and make the payment by credit card or paypal. Later he can come back here and check his order status.
 
Do you already have a steady and a provable income? Not just beer money but in thousands of $. If you have then you'll have a wider selection of banks to consider you, as many unfortunately do not open accounts for starting entrepreneurs.