You need a certain Pokemon.
Basically, each Pokemoon has a random, hidden number assigned to it by the game. It changes daily, and when you place it in the PC.
If the number your Pokemon has is a "Mirage Island" number, Mirage Island will appear for one day.
Unfortunately, there's no way to see or influence any of these hidden values, so it's easier to say that it's random and based on luck.