Impact of changing server machines names with SQL Server 2016 Standard in mirror state
Renaming Windows computer for SQL Server is supported an operation and described in documentation. For mirroring, it needs to be turned off and re-established after renaming.  Remember that renaming Windows requires a reboot, so be prepared for service outage.


  When you rename a computer that is configured to use database
  mirroring, you must turn off database mirroring before the renaming
  operation. Then, re-establish database mirroring with the new computer
  name. Metadata for database mirroring will not be updated
  automatically to reflect the new computer name. Use the following
  steps to update system metadata.


For what it's worth, could you use SQL client aliases instead of renaming the computers?