When you're hiking up the mountain in the blizzard you'll see a house with a Poke Ball behind it click it, and it will be hm08 better known as rock climb.
You do not need HM08 to reach Cinnibar Island. You do, however,
need HM08 to search Cinnibar Island. Without HM08, all you can do
is use the Pokemon Center.
Where do you get hidden move 08 in Pokemon LeafGreen?