Instructions
Travel to Pacifidlog Town. Walk to the house at the east end of the city and go inside.
Talk to the old man sitting by the window. He will tell you if he sees Mirage Island. If he does not see it (that is, if the five-digit numbers of your current Pokemon do not match the day's random five-digit number), exit his house and go to the Pokemon Center in the middle of the town. Go to the computer and switch out your current Pokemon party with others you have stored.
Go back to the old man's house and talk to him again. Continue switching Pokemon and talking to him until he either sees Mirage Island or you have held all of your Pokemon. In the latter case, wait until the next day (i.e., wait for the game to generate a new random five-digit code) and then start over.
Exit the old man's house if he tells you he can see Mirage Island. Go to the Pokemon Center and add a Pokemon with the Surf ability to your party if you do not already have one. Walk to the east end of Pacifidlog Town. Use "Surf" to cross the water and get to Mirage Island.