Personally I prefer NASA's pronunciation, which is "charon".

I was just being cheeky with the cha-cha-cha thing, but if I were to be a bit less useless more informative [sorry, I'll sperg out about etymology]:

[kʰ]→[x] likely started around 400 BCE. By then Aristophanes was representing Laconian/Spartan speech using ⟨σ⟩ instead of ⟨θ⟩, indicating that Laconian Greek underwent [tʰ]→[θ]; and since this change goes side-to-side with [pʰ kʰ]→[ɸ x], they're likely from the same times.

However odds are that the "old" sound [kʰ] survived for a few centuries in Attic and/or Ionic, as Latin still borrows a fair chunk of Greek words with ⟨φ θ χ⟩ being transliterated with ⟨ph th ch⟩, instead of ⟨f t~s h⟩ - even words related to Christianity like ⟨eucharistia⟩ or ⟨blasphemo⟩. The borrowing ⟨Charon⟩ is probably a bit older, so it's safe to say that you were "expected" to pronounce it with [kʰ] in Latin (...and in practice everyone adapted to [k] as it was the closest in the local phonology).

Modern Greek keeps that [x] for that word - [ç] only appears before [e i].

Then there's [ʃ] reading from the name Charlene, mentioned in the link. The name is surprisingly recent, Wiktionary claims that it's from the XIX century. By then French already underwent [tʃ]→[ʃ], although the digraph ⟨ch⟩ was likely repurposed by Old French to be used with [tʃ].