Instructions
Turn on your Wii. Press the "A" button on your Wiimote to bypass the Wii safety warning and load the main menu.
Select the Wii Message Board icon on the main menu. This is the circle with a picture of an envelope at the bottom-right corner of the menu.
Select the "Create message" icon. This is the circle with the pencil and paper directly to the left of the date display.
Select the Address Book, which is the right-most icon. Your Wii Number will be displayed on this screen, along with any friends you have connected with.