A good number of the languages in Anatolia weren't actually Anatolian, though, correct? Some of the major ones like Phrygian aren't Anatolian at all, and we have pretty scant evidence regarding a lot of the languages.
Seems like the key is really to maintain an indigenous, Anatolian state of sufficient power and cultural prestige that it doesn't find itself adopting Greek or getting conquered by an outside power. You really need a sort of dynastic cycle, where maybe individual regimes fall, but there's always one Anatolian-speaking empire waiting in the wings. It seems tricky, given how often Anatolia is invaded historically, but maybe not impossible. Maybe the Kashka (?) establish a second Hittite style kingdom in the fullness of time, assimilating into the broader Anatolian culture and ruling for a few centuries longer, before another dynasty overthrows them. Thus by the time the alt-Greeks are pressing on their shores, there's a long-standing power that isn't easily brushed aside with whom the Greeks have to treat as equals, a la the Persian Empire but more compact and regional. This averts the development of Aramaic as a lingua franca in the near east, and limits the incursion of Greek languages into Anatolia.
Keeping them around forever, I admit, is a little trickier. Sooner or later, as time goes on, it seems like the chaotic linguistic patchwork of Europe and the Middle East gets homogenized down, paralleling the establishment of imperial states and more developed trade routes. The Mediterranean world basically goes from dozens and dozens of local languages to Latin's successors, Arabic and related dialects, and Greek. It goes from loads and loads of small states to a series of empires and kingdoms that at least notionally claim to rule vast territories. At this point, you're basically relying on one of those regional empires being an Anatolian speaking one, some continuation state of that ancient Kashka successor, even if they have created their own, completely separate ideological foundation myth by now (which is likely). Ideally, the Anatolian state has also done some significant conquests - down into Syria, meddled in Greek politics a lot, that sort of thing. And hopefully it either has its own distinct religion or has adopted a religion and made it its own, using its own native tongue as the liturgical language in either case. That will all help it endure.
Then it just needs to survive a couple thousand years and pray nobody, Saka or Alan or Magyar or Turk or Mongol, doesn't come screaming out of the steppe and smash all of its progress, or that there's no great Iranian empire that conquers the whole near east, or that it doesn't fall and get replaced by a Greek empire, or a hundred other things don't go wrong.