Instructions:
Go to your Meta Business Manager Settings
Navigate to Accounts and click on WhatsApp Accounts and then (having selected the WhatsApp account you want to edit) click on WhatsApp Manager.
Click on Account tools and then Phone numbers on the left hand side and select the desired number.
In the ensuing window, under Display name press Edit.
Then, simply enter your new display name and click Submit.
Note: It can take up to 48 hours until Meta gives feedback on the newly submitted name. However, normally it only takes a few hours.