There is a story that explains that, and it is the oldest thing anybody has recorded about Serifos.
The chest
A king in Argos was told that his daughter's son would kill him. His daughter Danaë had a son anyway. So he put the two of them in a wooden chest and pushed it out to sea.
Putting his daughter with the child in a chest, he cast it into the sea. The chest was washed ashore on Seriphus, and Dictys took up the boy and reared him.
That is Apollodorus, in Frazer's translation of 1921. A fisherman called Dictys draws the chest out of the water. The boy is Perseus, and he grows up here.
It is worth knowing what Apollodorus is, because it is a good story about the story. He is a compiler, writing a long time after the fact, and the version we read is a summary of older accounts that have mostly not survived. Frazer traces it back through Pherecydes, quoted by scholars annotating other people's poems — a note on Apollonius, a note on Homer, a proverb collection, a Byzantine commentary. Sophocles wrote a Danaë and so did Euripides. Both are lost. What came through whole was the summary, and everything it summarised is gone.
So when anyone says the myth says, what they usually mean is that a late encyclopaedist wrote it down and the plays did not make it.
The return
Perseus grows up on the island. The king, Polydectes, wants Danaë and wants the son out of the way, so he sends him after the head of the Gorgon — an errand designed not to be survivable. Perseus survives it and comes back.
Having come to Seriphus he found that his mother and Dictys had taken refuge at the altars on account of the violence of Polydectes; so he entered the palace, where Polydectes had gathered his friends, and with averted face he showed the Gorgon's head; and all who beheld it were turned to stone, each in the attitude which he happened to have struck.
Read that carefully and it is a room. A palace, a gathering, a man walking in with his face turned away. Everyone who was in there.
And then Strabo
Strabo is a separate witness, writing geography rather than mythology, and he tells the same episode with one difference that changes its size:
And there is Seriphos, the scene of the mythical story of Dictys, who with his net drew to land the chest in which were enclosed Perseus and his mother Danaë, who had been sunk in the sea by Acrisius the father of Danaë; for Perseus was reared there, it is said, and when he brought the Gorgon's head there, he showed it to the Seriphians and turned them all into stone.
Not the friends in the palace. The Seriphians. He showed it to the island.
Which is the sentence to carry around with you here, because the ground agrees with it. Serifos is a granite island whose slopes are covered in standing and fallen stone, and the oldest surviving explanation of that is that the stones are people, caught in whatever attitude they happened to be holding when they looked up.
Apollodorus finishes the story kindly. Dictys, the fisherman who pulled the chest out of the sea, is made king of the island. The man who did the one decent thing at the beginning gets it at the end.
What the island says about itself
Two things, and they are worth keeping separate from the above.
The archaeological collection at Livadi is housed in the former Municipal Hostel Perseus. The island names its buildings after him, which is a quieter kind of evidence than a text — a place that has decided who it belongs to.
Serifos also tells a story connecting the island to Odysseus and the Cyclops. No ancient text puts that episode here. It is something Serifos says about itself rather than something Homer says, and that is genuinely interesting on its own terms: an island with one very well-attested myth has quietly acquired a second one. Enjoy it as local identity. Do not repeat it as classical.
The coins, and where they are not
Serifos struck silver from about 530 BC carrying a frog. Later island coinage carries Perseus himself, in a winged helmet, with the Gorgon's head and the curved sword.
They are not in the collection at Livadi. That is worth saying plainly, because it is exactly the mistake this essay could cause. The collection holds sculpture, architectural pieces, funerary reliefs and amphorae, from the Castle of Chora and Aspros Pyrgos, Classical through Roman. No coins. If you want to see a Serifos frog, you are looking for a museum somewhere else.
The collection is worth an hour anyway, and the practical position is narrow. Five euros. Half past eight to half past three, and it is shut on Tuesdays, which does not affect you. Wednesday the sixteenth and Thursday the seventeenth are your two days. On Friday it opens in the afternoon, and by then you are on the boat to Sifnos.
Why this one suits a walk
Because none of that is required.
There is no site to reach, nothing to queue for, no hour to get there by. The story's anchor is the ground under your feet, and it is the same anchor from every path on the island. Strabo says the Seriphians were turned to stone where they stood. You are walking through a hillside of stones that have stopped where they stopped.
That is not an argument that it happened. It is an observation about how good the story is: somebody looked at this island, saw what it is made of and how it lies, and produced an explanation that fits it exactly.