Trading
If you know someone who has a Lugia, you can easily add it to your Pokedex by simply trading with that person. Travel to any Pokemon Center in the Sinnoh region and talk to the trade attendant. Save your game and select "Trade" to begin. Have your friend select Lugia while you select any Pokemon you wish. Confirm the trade and, once the trade is complete, Lugia will be added to your party.
Navel Rock
Go to any Nintendo-sponsored Pokemon event to download the "Mystic Ticket" from a "WonderSpot." Once the ticket has been downloaded to your game via the "Mystery Gift" option, travel to the docks and show the sailor your "Mystic Ticket." He will take you to Navel Rock. At the top, you can encounter a level 70 Ho-oh. At the bottom floor of Navel Rock awaits Lugia, also at level 70. Be sure to bring your strongest Pokemon as Lugia is incredibly powerful and will not be captured without a fight.
Action Replay
With an Action Replay game modifying device, you can input a code that will allow you to battle Lugia in the wild. Simply enter the code "B749822B CE9BfAC1, A86CDBA5 19BA49B3, 65B97174 4A8FBA5C, 1CA7B8D8 B774D7CE" at the Action Replay code screen and load "Pokemon Emerald." Walk into any area where you can encounter wild Pokemon, such as a body of water, a cave or even a patch of grass. The only Pokemon you will battle will be level one Lugias. Capture as many as you wish and then save the game.
GameShark
As with the Action Replay, a GameShark device allows you to change the game to your liking using various codes. At the GameShark code screen, enter "A845D79CADEE, 866E1F5B8800, A845D79CADEE, 805C1D31C295" and load "Pokemon Emerald." Once you start the game, press and hold the "L," "B" and "Select" buttons to initiate the Lugia battle. Weaken and capture Lugia to add it to your party.