His character seems to have been chosen by Virgil treating the Persian-origin name Achaemenes as Greek and extracting the meaning "he who waits with affliction".
Although not mentioned in the Odyssey of Homer, which the Aeneid is a sequel to, Achaemenides is significant for being one of two known members of Odysseus' crew in literature to survive the return journey to Ithaca, along with Macareus (as every ship besides the flagship was destroyed by the Laestrygonian giants, and those besides Odysseus on the last ship were drowned after his men devoured Helios's sacred cattle).
The episode also provides Virgil with an opportunity to show Aeneas' magnanimity in saving a member of Odysseus' crew, and bearing no grudge for Odysseus' major role in the destruction of Troy, Aeneas' home.