It was mostly diaspora Jews who started using synagogues, and they started using them in a Hellenistic world. Greek was their primary language. They also used a Greek translation of the Hebrew Bible (called the Septuagint, often abbreviated as the LXX) After the destruction of the Temple, those Hellenistic Jewish communities in the Roman Empire became the core communities of Judaism and Greek was just the language that they already spoke.

There is a Hebrew term, by the way. Beit Knesset ("house of assembly").

The synagogue arose largely out of a Greek context. It became popular after the Temple was destroyed and the house, or a local community, became the focus. At this point in time, as the synagogue was becoming more ingrained in Judaism, the dominate language was Greek.

In Hebrew and Aramaic, there is a corresponding word to synagogue, beyt knesset. Some Jews also use the Yiddish term, Shul. And then other Jews use different words as well. Some even use the word kahal, which means community. So there has been other terms used throughout history and even today. But the main reason is because the synagogue as a proper structure, and a more important aspect in Judaism, rose during a time in which Greek was a dominate language. So it stuck.