Both Greek words, Hades and Tartarus do double-duty as both an abode for the dead, and a personified deity-figure, so the choice is not between naming a place after the god Hades and the use of Tartarus.

It's a little unclear whether hades first referred to the god or to the place, I have seen texts arguing for both derivations. Either way, the NT employs the term primarily to denote the place, and in translating the OT it stands for Sheol, similarily a dark shadowy place of the dead. Hades is used but a few times in the NT to personify death, but arguably not as a reference to the Greek god.

Why not Tartarus? The associations of Tartarus are different, it is a place below Hades, it is designated for the punsihmetn of the Wicked, and as a deity Tartarus is a primarodial force alongside Chaos and Earth. Within the Greek mythos, it came to be known as a place for the punishment of the Titans and a place to confine threats to the Olympian gods, so it's primary 'purpose' is not to hold the souls of dead human beings en masse, which is what Hades does. Only a verbal form of tartarō occurs in the NT, in 2 Pet 2:4, reasonably translated as 'thrown down to hell'. Alternatively, one should also look at the use of the word gehenna in the NT as a place of torment.

Of course, your question is actually a little backward, since the early Christians are talking about Hades, because they're functioning in Greek. Hell as an english word derives from the Old English Hel, with similar cognates across Germanic languages, which also seems to do double-duty as the place of the dead as well as a being ruling over the same place.