Admete, daughter of King Eurystheus
Admete (or Admeta) is the daughter of Eurystheus and Antimache. Heracles was obliged by her father King Eurystheus to fetch for her the girdle of Ares, which was worn by Hippolyte, queen of the Amazons. (Apollod. 2.5.9.) According to Tzetzes (ad Lycophr. 1327), she accompanied Heracles on this expedition.

There was a tradition (Athen. 15.672), according to which Admete was originally a priestess of Hera at Argos, but fled with the image of the goddess to Samos. Pirates were engaged by the Argives, inhabitants of Argos, to fetch the image back, but the enterprise did not succeed, for the ship when laden with the image could not be made to move.
The men then took the image back to the coast of Samos and sailed away. When the Samians found it, they tied it to a tree, but Admete purified it and restored it to the temple of Samos. In commemoration of this event the Samians celebrated an annual festival called Tonca. This story seems to be an invention of the Argives, by which they intended to prove that the worship of Hera in their place was older than in Samos.
Admete’s story, according to Athenaeus in The Deipnosophists. Or Banquet Of The Learned Of Athenaeus, Book 15, page 447
Admete, the wife of Eurystheus, after she had fled from Argos, came to Samos, and there, when a vision of Juno had appeared to her, she wishing to give the goddess a reward because she had arrived in Samos from her own home in safety, undertook the care of the temple, which exists even to this day, and which had been originally built by the Leleges and the Nymphs.
But the Argives (people from Argos) hearing of this, and being indignant at it, persuaded the Tyrrhenians by a promise of money, to employ piratical force and to carry off the statue of Hera —the Argives believing that if this were done Admete would be treated with every possible severity by the inhabitants of Samos.
Accordingly the Tyrrhenians came to the port of Juno, and having disembarked, immediately applied themselves to the performance of their undertaking. And as the temple was at that time without any doors, they quickly carried off the statue, and bore it down to the seaside, and put it on board their vessel.
And when they had loosed their cables and weighed anchor, they rowed as fast as they could, but were unable to make any progress. And then, thinking that this was owing to divine interposition, they took the statue out of the ship again and put it on the shore; and having made some sacrificial cakes, and offered them to it, they departed in great fear.
But when, the first thing in the morning, Admete gave notice that the statue had disappeared, and a search was made for it, those who were seeking it found it on the shore. And they, like Carian barbarians, as they were, thinking that the statue had run away of its own accord, bound it to a fence made of osiers, and took all the longest branches on each side and twined them round the body of the statue, so as to envelop it all round.
But Admete released the statue from these bonds, and purified it, and placed it again on its pedestal, as it had stood before. And on this account once every year, since that time, the statue is carried down to the shore and hidden, and cakes are offered to it: and the festival is called τονεὺς, because it happened that the statue was bound tightly (συντόνως) by those who made the first search for it.
Admete’s story, according to Apollodorus, in The Library, 2.5.9
The ninth labour he enjoined on Hercules was to bring the belt of Hippolyte.1 She was queen of the Amazons, who dwelt about the river Thermodon, a people great in war; for they cultivated the manly virtues, and if ever they gave birth to children through intercourse with the other sex, they reared the females; and they pinched off the right breasts that they might not be trammelled by them in throwing the javelin, but they kept the left breasts, that they might suckle. Now Hippolyte had the belt of Ares in token of her superiority to all the rest. Hercules was sent to fetch this belt because Admete, daughter of Eurystheus, desired to get it.
Admete as an Oceanid Nymph
A daugter of Oceanus and Thetys (Hesiod. Theog. 349), whom Hyginus in the preface to his fables calls Admlleto and a daughter of Pontus and Thalassa.
Hesiod, Theogony, 349:
And she [Titan Goddess Tethys] bore [with the Titan Oceanus] the holy family of Nymphs, who nurture men on earth with the lord Apollo and the Rivers, having this function allotted by Zeus:
Peitho and Admete, Vianthe and Electra,
Doris, Prymno, and godlike Urania,
Hippo, Clymene, Rhodea, and Callirhoe,
Zeuxo, Clytia, Idyia, and Pasithoe,
Plexaura and Galaxaura, lovely Dione,
Melobosis, Thoe, and fair Polydora,
Cerceis of lovely form, Pluto of big dark eyes,
Perseis, laneira, Acaste, and Xanthe,
lovely Petraea, Menestho, Europa,
Metis, Eurynome, and saffron-robed Telesto,
Chryseis, Asia, and desirable Calypso,
Eudora, Tyche, Amphirho, Ocyrhoe,
and Styx, who is chief among them all.
These were the eldest daughters born of Oceanus and Tethys; but there are many others too. For there are three thousand graceful-ankled Oceanids; widely scattered they haunt the earth and the depths of the waters everywhere alike, shining goddess-children. And there are as many again of the Rivers that flow with splashing sound, sons of Oceanus that lady Tethys bore. It is hard for a mortal man to tell the names of them all, but each of those peoples knows them that live near them.
Carl Kerényi, The Gods of the Greeks, page 41:
But most of the names of the daughters of Tethys I have yet to tell, and these are amongst the most puzzling: Peitho, Admete, lanthe, Elektra, Hippo, Klymene, Rhodeia, Zeuxo, Klytia, Idyia, Pasithoe, Melobosis, Kerkeis, Ianeira, Akaste, Xanthe, Menestho, Telcesto—she of the saffron-yellow raiment —and lastly Chryseis, Asia and Tychce.
One could make many conjectures concerning the goddesses hidden behind these names; but I shall make only the most obvious interpretations. Peitho, the goddess “Persuasion”, was clearly only one particular name of the Goddess of Love, and she therefore became a companion of Aphrodite.
Admete, on the other hand, was, like Artemis, an “untamable” and represented unwedded maidens while her sister Zeuxo represented the yoke of marriage.
Hippo and had to do with horse and cart. Idyia was a goddess of magical knowledge, Xanthe a fair-haired goddess, Telesto a goddess of initiations into mysteries, and Tyche was a goddess whose name means “what may hap” or “Chance”, a deity of whom no particular stories are told, but whose power —like that of the three Moirai and the threefold Hekate—proved stronger than the rule of Zeus.
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