There are two women of this name in ancient Greek mythology, unless somehow they are really one and the same.
The first is a daughter of Minyas of Orchomenos in Boiotia, wife of Iasos and by him mother of Meleagros’ beloved Atalante, but also by Helios of the unfortunate youth Phaethon. When the boy grew up, he was taunted with not having a father, and applied to his mother for information. She assured him that he was the son of none other than the Sun, and he set out to find his father. After long wanderings he reached the distant East where Helios’ palace is. Here he was received and recognized by Helios, who invited him to choose what gift he liked. He asked to be allowed to drive the chariot of the sun for one day. Helios protested vainly, and Phaethon next day mounted the car and essayed to guide the immortal horses. But he was unequal to the task, his team bolted, and Earth (Gaia) was in danger of being burned up. At last Zeus, at the appeal of Gaia, slew Phaethon with a thunderbolt, and Helios resumed charge of the fiery chariot. Phaethon fell into the Eridanos River, and his sisters wept for him till they were turned into trees from which distills the gum known to us as amber.
The other Klymene was a daughter of Minos of Crete’s descendant Katreus, and as usual with the women whose souls Odysseus conversed with in the place of the dead, her importance too lay in her offspring. This Klymene was the wife of Nauplios, and by him mother of one Palamedes, who not only went with Agamemnon to Troy but also helped recruit the other Achaian leaders for the expedition, and in so doing fell foul of Odysseus in the following manner:
Palamedes, son of Nauplios, was Odysseus’ rival in cunning; he was credited with having invented several letters of the alphabet, together with other ingenuities. When the muster for the Trojan war was forward, Odysseus pretended to be mad, so as to avoid joining Agamemnon’s cause. He therefore yoked a horse (or an ass) and an ox to his plough and proceeded to plough with them, sowing salt as he went; but Palamedes put Odysseus’ infant son Telemachos in the way, and noted that Odysseus was sane enough to turn the plough aside and avoid him; or he drew his sword against the child. Odysseus was henceforth his enemy.
Apollodoros tells in his Epitome how Odysseus repaid Palamedes for involving him in a ruinous war at the farthest limit of the Achaian world:
Having taken a Phrygian prisoner in the fighting, Odysseus compelled him to write a letter of treasonable purport ostensibly sent by Priam to Palamedes; and having buried gold in the quarters of Palamedes, he dropped the letter in the camp. Agamemnon read the letter, found the gold, and delivered up Palamedes to the allies to be stoned as a traitor...
...And when Nauplios learned of it, he sailed to the Greeks and demanded satisfaction for the death of his son; but when he returned unsuccessful (for they all favored King Agamemnon, who had been the accomplice of Odysseus in the murder of Palamedes), he coasted along the Grecian lands and contrived that the wives of the Greeks should play their husbands false, Klytaimestra with Aigisthos, Aigialeia with Kometes son of Sthenelos, and Mede, wife of Idomeneus, with Leukos. But Leukos killed her, together with her daughter Klisithyra, who had taken refuge in the temple; and having detached ten cities from Crete he made himself tyrant of them; and when after the Trojan War Idomeneus landed in Crete, Leukos drove him out. These were the earlier contrivances of Nauplios; but afterwards, when he learned that the Greeks were on their way home to their native countries, he kindled a beacon fire on Mount Kaphereus (which is now called Xylophagos); and there the Greeks, standing in shore by night in the belief that it was a harbor, were cast away on the rocks.