The 2nd-century BC travelogue known as the Description of Greece describes the origin of the name of a certain town, which the travelogue’s writer Pausanias says came from Tanagra, a daughter of the Boiotian river Asopos, who lived in that town. As she approached an extreme old age, so the story went, Tanagra’s neighbours nicknamed her Graia, “Old Woman” (literally “Grey”), which name was eventually also given to the town.
Fibi, the dude you're thinking about is definitely Tithonos [Tithonus], who, by the way, was a brother of Priamos [Priam], who succeeded their father Laomedon as king of Troy, and thus was the ruler of the city during the Trojan War. Tithonos and Eos were the parents of Emathion, who became king of Egypt and Arabia, and of Memnon, who became king of Ethiopia, and supported his uncle Priamos and the city of Troy during the war. Emathion was killed in Libya by Herakles [Hercules] while Memnon was slain a generation later by Achilles in the Troad.
The Trojan War actually features a bunch of really old people, including the Trojan king Priamos himself, who by the time his kingdom was being besieged and destroyed by the Greeks was a frail, white-haired dude. His queen Hekabe [Hecuba] was similarly elderly at this point, as were the Elders of the royal court, among whom were Priamos’ older brothers Klytios, Lampos and Hiketaon, as well as their father’s cousin Antenor and his wife Theano (who was a sister of Hekabe). Even Hiketaon’s son Thymoites was old enough to be one of the city Elders by that point in time. Thymoites was married to his own aunt Killa (a daughter of Laomedon and sister of Hiketaon and Priamos), who I imagine would likewise have been old by this point. Prince Ankhises [Anchises] of Dardania, a kingdom neighbouring Troy, was a second cousin of Priamos, and his claim to fame is that he was the father, by the goddess Aphrodite, of Aeneas, whose descendants founded Rome. Ankhises was old and blind during the Greek invasion of Troy and so Aeneas had to carry him on his shoulders during their escape as the city was being razed.
Other old people of the Trojan War include Oukalegon [Ucalegon], another Trojan Elder; Prince Phoinix [Phoenix] of Ormenion, who helped to raise Achilles; Briseus and Khryses, the priests whose daughters, respectively Hippodameia Briseis and Astynome Khryseis, indirectly caused a row between Agamemnon and Achilles (which eventually resulted in the death of Achilles’ best friend Patroklos). King Oineus [Oeneus] of Aitolia was also very old during this war, as was Lairtes [Laertes], the father of Odysseus.
Some of the immortals were portrayed as aged beings, such as the Centaur Kheiron [Chiron]; the minor rustic god Seilenos; Kharon [Charon], the rugged grey boatman of the Underworld; the sea-gods known as the Old Men of the Sea, namely Nereus (father of the Nereides), his two younger brothers Phorkys (the father of many monsters, most famously the Gorgons and the Hydra) and Thaumas (the father of Iris and the Harpies), as well as the minor sea-god Proteus; the frail elderly goddesses called the Litai [Litae], who personified prayers of repentance; and Geras, the personification of old age itself, who was depicted in art as a tiny, shrivelled-up old man. Aion, the personification of eternity, was a form of the Orphic deity Khronos [Chronus, “Time”], who eventually developed in modern times into Father Time or “Old-Man Time,” who is represented as an old man with a long white beard and long white hair. This representation dates back to the ancient Egyptian poet Nonnos, who described him this way. From Aion we get the English word aeon [or eon].
The most notable [immortal] old women of Greek myth have got to be the triplet Graiai, “Grey Ones,” sisters of the three Gorgons, who were born wrinkled and grey-haired, with only one eye and one tooth to share among themselves. Ancient Greek and Roman writers also tell of an Indian tribe, whom the Greeks called the Mandoi (which in Latin is Mandi), dwelling in the foothills of Mt Himalaya, whose lifespan was so short that they grew old and wrinkled almost as soon as they were born.