How can I assign a customisable mouse button to open Notification Center?
Depending on your mouse and driver software, your best bet would probably be to set a keyboard shortcut for Notification Center and program your mouse button to trigger that shortcut.

You can set up the shortcut in System Preferences -> Keyboard -> Keyboard Shortcuts -> Show Notification Center.


The setup for your mouse driver will vary, but on a Razer mouse you can assign the button to a Keyboard Function and type your shortcut.