Instructions
Visit a Pokemon Center and manage your Pokemon team. You will need a Pokemon that knows the Surf ability, the Whirlpool ability and the Flash ability.
Fly to Olivine City and travel east to Route 40. Head south to Route 41 by using the Surf ability in the water.
Swim to the northwest island in Route 41. When you swim up to a whirlpool, you will be asked to use the Whirlpool ability to cross. Follow the path to reach the northeast island.
Enter the cave from the northeast island. Open your menu to select the Flash ability in order to produce light in the cave.
Walk east and jump down from the ledge. Jump down the northernmost ledge and exit using the ladder to the north.
Follow the path south, then west, then south again to reach the next ladder. Take the ladder to enter the lower portion of the cave.
Walk south and talk to the old man. If you have the Silver Wing he will let you pass. Walk down the path to enter Lugia's chambers.
Talk to the Kimono Girls in Lugia's chamber. After a short cut-scene they will summon the Legendary Pokemon Lugia.