Piccolo Day is held on May 9 (May 13 in the dub). This is the day that King Piccolo declares his ruling of Earth.
The numbers 5 and 9 can be read in Japanese as "go" and "ku", respectively, this is presumably why Akira Toriyama picked this date specifically.
In 2015, the real-world Japan Anniversary Association recognized on May 9 as Goku Day because of the aforementioned reason.[1]
Additionally nine is considered to be unlucky because the word for nine (ku) sounds like the word for "pain/distress" (苦, ku). In English-speaking cultures, 13 is considered unlucky, which is likely why the dub changed the date to the 13th.