In the same rooms where you have to complete a picture puzzle of Omanyte, Kabuto, Aerodactyl, or Ho-Oh, there is a plaque that has a word that is spelled using Unowns. Do the following steps.
Room with the Kabuto Puzzle The word is "Escape". You must use an Escape Rope facing the plaque. You can easily get an Escape Rope at the Violet City Pokemart. The four items obtained are Energy Powder, Heal Powder, Oran Berry, and Pecha Berry.
Room with the Aerodactyl Puzzle (accessible by Surfing) The word is "Light". Use Flash while facing the plaque. The four items obtained are Energy Root, Heal Powder, Moon Stone, and Sitrus Berry.
Room with the Ho-Oh Puzzle The word is "Ho-Oh". You must have Ho-Oh first in your roster. The items obtained are: Charcoal, Leppa Berry, Life Orb, and Revival Herb. In Pokemon HeartGold, Ho-Oh is obtained in the Bell Tower. In Pokemon SoulSilver, Ho-Oh is obtained after you have defeated the Elite Four.
Room with the Omanyte Puzzle The word is "Water". You must have a Water Stone in your bag. The items obtained are: Leppa Berry, Mystic Water, Stardust, and Star Piece. A Water Stone can be obtained by winning it at the Pokeathlon on Wednesday.