When Zeus has stolen the 5 sacred items, get Hercules to help you get the Trident and the Hades Crown, then battle Zeus on Mount Olympus. When you win, Athena gives you the Island Medallion.
In More Detail
After you get the items on the scroll Zeus gives you, you show them to Athena and then Zeus appears and takes the items. Athena tells you that her brother Hercules will help and he does. After you get items from Poseidon and Hades, go to the gate to Mount Olympus with Hercules and he will open the gate for you. Your character continues after Hercules becomes a statue, and you make your way to the top of Mount Olympus and challenge Zeus. Once you win, he will disappear and you get the medallion from Athena.
(see related solution questions below)
After you pass the gates you will have to battle Zeus and after that you will win the island medallion from his daughter.
On Mythology Island in Poptropica, the final thing you do is defeat Zeus in a battle to stop him from taking over the island. Once you defeat Zeus, you will have completed the main quest of the island and can claim your medallion.
That completes the Mythology Island quest, and Athena gives you the island Medallion. You can leave Olympus and go to other islands. Wearing the Hades Crown on any island lets you grow into a giant (press the spacebar). But it does provide any advantage on any island.
You need the 5 sacred items to defeat Zeus and the island to get the medallion
Zeus must be sulking, because he is nowhere to be found. Athena gives you the Island Medallion and you can go down from Olympus to leave the island. Ominously, Zeus turns up again on a later Poptropica island.
Beating Zeus wins the game. You will automatically return to normal and give the 5 items to Athena for safe-keeping. She gives you the Island Medallion and you leave the island. Zeus is not seen again on the island. (see related question)
Zeus is the god on Mythology Island.
After you travel back to the future to get the Island Medallion, return to the present to leave the island (just as you did after completing the items), by using the Lab selection on the time device.
you leave and go to a different island
You can go to another island and complete it. When you get a medallion you have completed that island :)
You get the Island Medallion after you prove the existence of three of the four cryptids, and rescue Bigfoot from the evil Gretchen Grimlock. (see related question)
From the museum in Mythology island.