Ah. Good question.
There are four entrances to the Whirl Islands cave/"dungeon," but only one will get easily get you to the deepest part of the cave, in which Lugia dwells. (And another entrance will, in fact, get you nowhere because it is an instant dead end!) The correct entrance is on the upper-left island.
Make sure you have the Silver Wing from the Goldenrod Magnet Train Station manager (after driving Team Rocket out of Goldenrod City) before venturing into Whirl Islands, or Lugia will not be there.
all of the island
Whirl islands
HM07 Waterfall is found in the Ice Path, not the Whirl Islands.
I believe the Top Right island.
By fishing or surfing in the whirl island caves
It's in the third island.
you go to whirl island and go in a cave then down a waterfall and into a room.
go to whirl island and talk to the man on the third floor down then you can go past him
whirl islands is in jonto and you can get there by surfing south from olivine city. you can get a lugia in there if you have a silver wing. i hope you find your way through! -RR14 :)
get the silver wing from an old man in pewter city,go to whirl island,top right
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.)
The Silver Wing in Pokemon Silver lets you interact with Lugia in the Whirl Islands to fight and catch it. Lugia can be found in the deepest part of the Whirl Islands.