If I have Physical Double face down on the field and my opponent attacks with a monster on his side of the field, then I flip Physical Double after the attack, does his attack hit my mirage token or does he get to choose to attack again?

Whenever a player attacks, they must choose a particular monster or (when possible) a player to receive the attack.

Whenever the opponent changes the number of monsters on their field, a "replay" will occur. A replay will allow the turn player to choose a new target for the attack if they so choose.

In your case, by putting that token into play the opponent will now have to choose what they want to attack now (or IF they want to attack at all).

Magus, you've got it correctly; however, a replay is also triggered when the number of attack targets changes.

Don't forget that only happens during the battle step . . .