It's a program with no blocking I/O, so if the eyes stop moving, you know you're looking at a certain kind of connectivity issue. If some more complicated app stops responding, the problem space of where to start troubleshooting is much larger.

If you miiiiight have a flaky network, leaving xeyes open in the corner can be a real headache-saver.

I use xeyes as a kind of "is remote X working" test. Quick way to test that X forwarding through ssh is functional and fast enough.

I use it to check if an app uses xwayland when I use sway