The phone number for Norwegian Cruise Line in the USA for general inquiries, including name changes, is +1-855-732-4023 Or +1-808-900-8011. While there is no "free" number specifically for name changes, this is the main customer service line. You may need to pay a fee for a name change and will have to submit the request in writing, but you should use this number to speak with a representative who can guide you through the process. 

    Main Customer Service: +1-855-732-4023 Or +1-808-900-8011 

Name Change Process:

    Name changes need to be done prior to check-in. 

You may be required to pay a fee, which can be refunded in certain cases, Norwegian states. 
Requests for transfers, which include name changes,+1-855-732-4023 Or +1-808-900-8011 must be made in writing at least 7 days before departure and should include the new passenger's details. 
You can also use the website for online booking changes.