Returning from exile in the 5th century BCE, Jews brought with them the Babylonian names for the months and the Babylonian calendar itself. As Sacha Stern writes, it suited them:
The Babylonian calendar originated in Babylonia (southern Iraq) in the early second millennium BCE, spread to the rest of Mesopotamia in the late second millennium BCE, and then became, in the first millennium BCE, the official calendar of the great empires of Assyria, Babylonia, and Persia, in use across the whole Near East. The Jews under Persian rule adopted it as their own calendar, as did many other peoples in the Persian empire.
The Jews adopted not only Babylonian month names but also the entire Babylonian calendar. This calendar was lunar, with each month beginning at the sight of a new moon. Since twelve lunar months are approximately eleven days shorter than the solar year, the Babylonian calendar was intercalated (or evened out) every two or three years by the addition of a thirteenth month (usually by duplicating the twelfth month, Adar, and less frequently by duplicating the sixth month, Elul). This allowed the lunar system to catch up with the sun and the seasons. This calendar may have been quite similar to the original Israelite one, which was most likely also lunar; indeed, this may have helped the Jews to adopt it without qualms. . . .
[A]fter the Jewish Hasmonean state broke off from its Hellenistic Seleucid overlords in the mid-2nd century BCE, the Jews no longer had any reason to comply with the calendar of distant Babylon, and their calendar soon acquired distinct features.