It's a phrase used to celebrate the day honoring the saint a person is named after. It's similar to celebrating a birthday, but instead of celebrating the day you were born, it celebrates the day of your patron saint.
Here's a breakdown:
* Feliz: Happy
* Día: Day
* de: of
* tu: your
* Santo: Saint
So, when someone says "Feliz Día de tu Santo," they are wishing you a happy day in honor of your patron saint.