Unfortunately, the short answer is that you need to enter their Friend Code, something that Nintendo now calls the Wii Number.

Here's how it works:

The console, as I pointed out in a video, has an address book. In the address book you can find out your console's Wii Number, register a friend or send a message.

To register a friend you have to be connected to the Internet. Next you tap in their nickname then enter the Wii Number for their console or, if your friend is on a PC or cell phone, you enter their email address.

Keep in mind that to register another person as a Wii Friend, both of you need to register each other.