The cave took its name from Philoctetes who lived here, when after a snake bite he was abandoned in Lemnos island from Odysseus, so that the Achaeans could continue their journey to Troy.
Story of Philoctetes: Philoctetes was, according to Greek mythology, the son of King Poeas of Meliboea in Thessaly. He was a Greek hero, famed as an archer, and was a participant in the Trojan War. He was the subject of at least two plays by Sophocles, one of which is named after him, and one each by both Aeschylus and Euripides. However, only one Sophoclean play survives—Aeschylus' Philoctetes, Euripides' Philoctetes and Sophocles' Philoctetes at Troy are all lost except for some fragments.
Hercules wore the shirt of Nessus and built his funeral pyre, no one would light it for him except for Philoctetes or in other versions his father Poeas. This gained him the favor of the newly deified Heracles. Because of this, Philoctetes or Poeas is given Hercules' bow and poisoned arrows.
Philoctetes was one of the many eligible Greeks who competed for the hand of Helen, the Spartan princess; according to legend, she was the most beautiful woman in the world. As such, he was required to participate in the conflict to reclaim her for Menelaus in the Trojan War. Philoctetes was stranded on the island of Lemnos by the Greeks on the way to Troy. There are at least four separate tales about what happened to strand Philoctetes on his journey to Troy, but all indicate that he received a wound on his foot that festered and had a terrible smell. One version holds that Philoctetes was bitten by a snake that Hera sent to molest him as punishment for his or his father's service to Hercules.
Helenus, the prophetic son of King Priam of Troy, was forced to reveal, under torture, that one of the conditions of the Greeks' winning the war was that they needed the bow and arrows of Hercules. Upon hearing this, Odysseus and a group of men rushed back to Lemnos to recover Heracles' weapons. Surprised to find the archer alive, the Greeks balked on what to do next. Odysseus tricked the weaponry away from Philoctetes, but Diomedes refused to take the weapons without the man. Hercules, who had become a god many years earlier, came down from Olympus and told Philoctetes to go and that he would be healed by the son of Asclepius and win great honor as a hero of the Achaean army.
Once back in military company outside Troy, they employed either Machaon the surgeon or more likely Podalirius the physician, both sons of the immortal physician Asclepius, to heal his wound permanently. Philoctetes challenged and would have killed Paris, son of Priam, in single combat were it not for the debates over future Greek strategy. In one telling it was Philoctetes who killed Paris, he fired four times, the first arrow went wide, the second struck his bow hand, the third hit him in the right eye, the fourth hit him in the heel, there was no need of a fifth shot. Philoctetes sided with Neoptolemus about continuing to try to storm the city. They were the only two to think so because they had not had the war-weariness of the prior ten years. Afterward, Philoctetes was among those chosen to hide inside the Trojan Horse, and during the sack of the city he killed many famed Trojans.
Source: wikipedia.org