There have actually been many names from almost as many sources. In the Aeneid, Virgil described the priestess as "Deiphobe, daughter of Glaucus":
ni iam praemissus Achtes
Adforet, atque una Phoebi Triviaeque sacerdos,
Deiphobe Glauci, fatur quae talia regi:
'Non hoc ista sibi tempus spectacula poscit;
'Nunc grege de intacto septem mactare iuvencos
'Praestiterit, totidem lectas de more bidentes.'
SOURCE: Aeneid Book 6, 36 ff (translation in endnote)
In the Description of Greece, Pausanias, citing a Cumaean historian, named her "Demo":
The next woman to give oracles in the same way, according to Hyperochus of Cumae, a historian, was called Demo, and came from Cumae in the territory of the Opici. The Cumaeans can point to no oracle given by this woman, but they show a small stone urn in a sanctuary of Apollo, in which they say are placed the bones of the Sibyl.
In the Etymologiae, Isidore of Seville claims the Cumaean oracle is called "Amalthaea":
Septima Cumana, nomine Amalthea, quae novem libros adtulit Tarquinio Prisco, in quibus erant decreta Romana conscripta. Ipsa est et Cumaea, de qua Vergilius
Isidore may have traced this name from the Antiquitates Rerum Humanarum et Divinarum, in which Varro reputedly gave the names for the Cumaean Sibyl: "Herophile" and "Demophile" in addition to Amalthaea. This work is now lost, but his Sibylline list is preserved for us by Lactantius in The Divine Institutes:
The seventh was of Cumæ, by name Amalthæa, who is termed by some Herophile, or Demophile, and they say that she brought nine books to the king Tarquinius Priscus, and asked for them three hundred philippics, and that the king refused so great a price, and derided the madness of the woman; that she, in the sight of the king, burnt three of the books, and demanded the same price for those which were left.
Martianus Capella however claims the Cumaean and Erthraean were one, "Symmachia":
You know that there were not ten of these sibyls, as they say, but two — namely, Herophila, a Trojan, daughter of Marmesus; and Symmachia daughter of Hippotensis, who was born at Erythrae but prophesied at Cumae
In the De Mirabilibus Auscultationibus, it is claimed that "some" called her "Melankraira":
At Cumae in Italy there is shown, it appears, a subterranean bed-chamber of the prophetic Sibyl, who, they say, was of a very great age, and had always remained a virgin, being a native of Erythrae, but by some of the inhabitants of Italy called a native of Cumae, and by some named Melancraera.
Lastly, the scholia on Plato's mentions that the Cumaean Sibyl has been called "Taraxandra".
The Cumaean sibyl had once the apotropaic name of Taraxandra, 'she who alarms men' (Schol. Plat. Phaedr. p. 244B), which suggests the severity of a guardian at a gate.
Knight, William Francis Jackson. Cumaean Gates: A Reference of the Sixth Aeneid to the Initiation Pattern. Blackwell, 1936.
Translation of Aeneid, Book 6, lines 36-40:
But now Achates comes, and by his side
Deiphobe, the Sibyl, Glaucus' child.
Thus to the prince she spoke :
“Is this thine hour
To stand and wonder? Rather go obtain
From young unbroken herd the bullocks seven,
And seven yearling ewes, our wonted way.”
Editor's Note: Phoebi Triviaeque sacerdos is translated simply as "the Sibyl" but the literal translation might be "the (sacred) priestess of the temple of Apollo at Nemi".
Triviaeque may have secondary allusion--a place where 3 roads meet, associated with both Hecate and Diana--but here specifically meaning "(the Lake of Diana), a lake in Latium, near Aricia, now Lago di Nemi"