How can I show a custom “Bye bye” message when Linux Mint shuts down?

Look for an existing Plymouth theme that uses text or simple graphics, then customize its shutdown portion rather than building everything from scratch. Linux Mint has different editions and desktop setups, so the exact theme location and commands can vary depending on whether your installation is Ubuntu-based or Debian-based.

Plymouth includes a text-based splash module with configurable messages, although its documentation is fairly limited. Searching for Plymouth theme examples and testing changes in a virtual machine or with a backup is safer than editing the system theme directly.

This is probably done through Plymouth, the system that displays the boot and shutdown splash screen. You could create or modify a Plymouth theme and replace the normal shutdown text or graphics with your own message. Keep in mind that shutdown happens quickly, so the message may only appear briefly, and some systems may power off before it can be shown for long.