After you rescue Luigi from the Ghostly Galaxy, he will occasionally leave the garage and send you letters. In these letters, there's a picture of him (Note: you can only obtain these Power Stars once Luigi has sent the letter). You'll have to determine the galaxy he's in, and then the level. Below are the three "Luigi Stars":
Luigi Star 1
You will get a letter from Luigi looking like this. He's in the Good Egg Galaxy. Enter the Dino Piranha level. Luigi is waiting for you on top of the little Mushroom House. The easiest way up is a Triple Jump. You can also find a Golde Pipe on the underside of the planet, which will take you to a room. After completing the room, you will come out on the roof. Either way, Luigi will be waiting with a Power Star.
Luigi Star 2
You will get a letter from Luigi looking like this. He's in the Battlerock Galaxy. Enter the Battlerock Barrage level. Proceed as normal. At the very end, run to the underside of the platform containing the caged Power Star. Here is a caged Luigi. Lure a Bullet Bill to the underside and destroy Luigi's cage. Luigi will give you a Green Star.
Luigi Star 3
You will get a letter from Luigi looking like this. He's in the Honeyhive Galaxy. Enter the Bee Mario Takes Flight level. Proceed as normal until you get to the tunnel behind the waterfall. Stand on the very edge, facing the waterfall, and fly to your left (Mario's right) or the left side of the screen. You should land on a grassy area with a tree containing your brother, Luigi. Shoot him with a Star Bit, or jump to him to knock him down. He will then give you a Power Star.