When re-purposing an iPhone, from the Verizon MyBusiness Portal Verizon Business Account Sign In | Verizon (verizonenterprise.com) tech with admin rights for Verizon will need to sign in, select Manage Account> Manage Wireless Numbers, search for the mobile number and EDIT User Information to change the name of the user to the new user.
This may or may not change the caller ID in a timely manner. If it fails to do so, call Verizon Wireless support at 800.922.0204 / technical support using pin number 1390 and the phone number of the line to be change. Request Caller ID change of the support representative.
24-48 hours may be required for the change to take place.