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

This is probably something you’d customize through Plymouth, the system that displays graphics and status messages during boot and shutdown. You could start with an existing Plymouth theme and modify its text or image to show your message. Keep in mind that shutdown can happen quickly, so the message might only appear briefly.

Plymouth includes a text-based module with configurable messages, although the documentation is fairly limited. A custom theme or script-based theme would likely be the most practical approach. Before changing anything, check whether your Mint installation is Ubuntu-based or Debian-based, since the exact theme files and update commands can differ.

You can use the normal startup and shutdown status messages as a starting point and replace or extend them with your own wording. It may also be possible to trigger a spoken message, but changing the shutdown screen itself is more likely to work reliably.