The first line of Homer's Odyssey contains an invocation to a goddess that the poet calls only "Muse" (Greek: mousa). Homer's Iliad also opens with an invocation to an unnamed divinity that the poet calls only "goddess" (Greek: thea).
Early in the Theogony, Hesiod, who lived around the same time as Homer, tells us that there were nine muses, who were the daughters of Zeus and Mnemosyne ("Memory"):
"These things, then, the Muses sang who dwell on Olympus, nine daughters begotten by great Zeus, Cleio and Euterpe, Thaleia, Melpomene and Terpsichore, and Erato and Polyhymnia and Urania and Calliope, who is the chiefest of them all, for she attends on worshipful princes" (Evelyn-White translation)
The final name is Hesiod's list, Calliope, is the Muse associated with epic poetry, so we might assume that Homer has in mind this divinity when the poet asks for inspiration for epic poems like the Iliad and Odyssey.