The question and the question details are asking different things, and I’ll address them separately.
It is the doom of /k/ in front of a front vowel (i, e) to be palatalised, to be pronounced as [kʲ] > [c]. The palate is a notoriously difficult place to articulate a stop (too much surface area). So [c] more often than not ends up become (a) an affricate and (b) palatoalveolar: [tʃ] (moving forward in the tongue root, to where there is a more well defined articulator). It can move even further, and become an alveolar affricate: [ts].
That’s what’s happened to Latin <ci> throughout Romance. Caelum would have started as [cielu]; then [tʃielo] in Italian, then [tsiel] in French, which then simplified further to [siel].
The same change has happened in the Cretan and Cypriot dialects of Greek: /kokina/ ‘red’ > [kotʃina]. Standard Greek, on the other hand, stops at [c]: [kocina]. Both Crete and Cyprus spent time ruled by Venetians. Did they get [tʃ] from the Venetians?
No need to: this is a linguistic commonplace, and it has happened in dialects of Greek with no contact with Italian: Tsakonian, for instance (/kairos/ > [tɕere]), or Cappadocian (/kelyfos/ > /tʃefos/). In fact, Peloponnesian–Heptanesian, the base dialects of Standard Modern Greek, are outliers in not having palatoalveolars.
The question details throw some words:
Latin Aretium > Italian Arezzo
Darıca, a town in Turkey whose Greek name was Aretsou, and whose ancient name may have been Arethusa (though its classicising name was Rhysion).
Now, alveolars also palatalise cross-linguistically: Latin <ti> has indeed been through a bunch of changes, moving it towards the roof of the mouth, although it typically does not go further than affrication: [ti] > [tsi]. So Latin natione > Italian nazione [natsione] > French [natsion] > [nasion]. English in turn palatalised [sj] to [ʃ].
The corresponding palatalisation of /ti/ in Greek is rarer, but it has happened, and when it has happened, it’s been spectacular: /ti/ goes all the way back to [ci], in Tsakonian and Lesbian.
So. What about Aretsou?
I found out only today that [θθ] > [ts] was a thing in Finnish: Joonas Vakkilainen’s answer to What did your language sound like 500 years ago? (thanks, Joonas). But I’ve gotta say, I’ve never seen the equivalent in Greek. There is what <θ> pronounced as <σ> in Laconian, which appears backed up as rare instances of /θ/ > /s/ in Tsakonian. But that change was ancient, and likelier Laconian pronouncing /tʰ/ as /θ/ much earlier.
Aretsou is in Bithynia, so I was going to say “forget it, Bithynia was resettled by Greek-speakers in the 16th century from Epirus, there can’t have been any continuity from ancient Arethusa”. (Bithynia was the Ottoman heartland, so it was Turkicised early.) But Darıca is only 40km from Istanbul, so it’s plausible that it remained Greek–speaking after the Ottoman conquest.
Still, because I haven’t seen [θ] > [ts] elsewhere in Greece, I think Arethusa > Aretsou is unlikely. Any connection with Arezzo is also unlikely.