lugia is on the top right island
On whirl island
At the bottom of WHIRL ISLAND.
Whirl islands
the top one on the right
You catch Lugia there.
Which island? whirl islands? Cinnabar island? you catch lugia in whirl islands. you beat blaine in cinnabar island.
whirl islands
the fourth one to the bottom right
lugia is in the whirl island on the top right cave
To find Lugia enter the Whirl Island Caves at the top-right island, then it should be pretty easy from there (if you're stuck use youtube or similar.)
In Pokemon Heartgold, Lugia can be found at the top-right island of the Whirl Islands Area. It's at Lv. 70 so prepare.
I believe the Top Right island.