Hesiod's Tartarus is a measurement. A bronze anvil would fall nine days from heaven to reach the earth, and nine more from the earth to reach Tartarus, and it is bounded by a bronze wall and by night poured round its neck. It is not described as a place of torment. It is a container, built to hold the Titans after the gods defeated them, with Hundred-Handers set as guards.
The punishment came with the named prisoners, and they became a canon: Tantalus with the water and the fruit that withdraw, Sisyphus and the stone, Ixion on the wheel, Tityos and the vultures, the daughters of Danaus filling a jar that will not hold water. What the list has in common is that every punishment is a task that cannot be completed. That is a specific idea about what the worst possible sentence is, and it is a more interesting one than fire.
Later writers, especially Virgil, gave Tartarus a judge, a gate, a road and a population of ordinary sinners, and Christian writing found the structure ready to use. The New Testament uses a verb formed from the name once. What began as a sealed vault for defeated gods had become, within a few centuries, a place where individuals are sent for what they did.
The word also became a verb of sorts in later usage, and the English adjective derived from it now means a state of misery rather than a place. That is the usual fate of these names: Hades, Gehenna and Tartarus all began as somewhere and ended as a condition.
Sources AeneidVirgil (Publius Vergilius Maro) with The New Testament, 2 PeterTraditionally attributed to Peter the Apostle