Final answer:
The Homeric Hymn to Demeter was probably composed during the late eighth to the third quarter of the fifth century BCE, with the latter half of the fifth century BCE being the most likely period of its composition.
Step-by-step explanation:
The Homeric Hymn to Demeter is a foundational myth central to the Eleusinian Mysteries and devoted to the goddess Demeter. The hymn narrates the story of Demeter's search for her daughter Persephone, who was abducted by Hades, and the subsequent agreement that led to the cycles of the seasons. Its composition is usually dated to the seventh or sixth century BCE, during the Homeric Age, which is the time when the epic poems of Homer, the Iliad and the Odyssey, were first recorded. Specifically, the literary and historical evidence suggests a narrower timeframe: the hymn is attributed by some scholars to around the late eighth century BCE, but most compellingly, the reference in the Derveni Papyrus to the Orphic Hymns, to which the Homeric Hymn to Demeter is related, suggests that it was known and attributed to the latter half of the fifth century BCE. This period, often related to the influence of the Sophist Prodicus, indicates that the Hymn's origin may well be around the third quarter of the fifth century BCE, situated in Attica or even Eleusis itself, making it the product of a mature phase of Athenian cultural expression.