You can add friends by sending an invite URL via e-mail or other means and then having them submit a friend request by doing the following:
After opening the Friend List, tap Invite to open the Invite Friend screen. Then select how you want to send the invite.
A message containing an invitation URL will be created, which you can send to your friend.
Once the user has accessed the link and followed the instructions for sending you a friend request in Super Mario Run, you'll be able to accept the request by first tapping Accept from your Friend List and then Approve.
Related question: "How do I add friends?"