The word kykeon means a stirred mixture and nothing more. Homer uses it for perfectly ordinary drinks: one is mixed for a wounded man in the Iliad, and Circe serves another in the Odyssey, into which she stirs her drug. Nothing about the term is sacred. What makes the Eleusinian kykeon matter is a single scene and the rite that scene explains.
In the Homeric Hymn to Demeter the goddess, searching for her daughter and disguised as an old woman, comes to Eleusis and sits in the house of Keleos refusing food and drink in her grief. She is offered wine and refuses it, saying it is not permitted to her, and asks instead for barley meal and water mixed with glechon, a mint. She is given it and drinks, and that ends her fast. Initiates at Eleusis drank a kykeon in their turn, and the hymn functions as the charter for that act: the initiate does what the goddess did, in the place where she did it.
Beyond this the record closes. The Mysteries were protected by an oath of silence that held for roughly a thousand years and was taken seriously enough that violating it was a capital matter. What the initiates saw, what was said and what was shown are not described by anyone who knew, and the ancient authors who mention the Mysteries at all are either outsiders or are careful to say nothing. The result is that the kykeon is one of the very few concrete details anyone has about the central rite of the Greek world, which is precisely why so much weight has been placed on it.
The drink was preceded by fasting, and the fast and the drinking were the frame of the initiation rather than incidental to it.