In the pre-Babylonian era, we find in the Scriptures only four months on the calendar that are identified by name:
The first month (Nissan): Aviv1
The second month (Iyar): Ziv2
The seventh month (Tishrei): Etanim3
The eight month (Cheshvan): Bul4
The other months were just known by their place in the calendar—e.g., third month, fourth month—starting from the first month: first by virtue of the fact that it is the month when our nation left Egypt, the month when we became a nation.
(Apparently, even the four months that had names, were more often than not referred to by their numeric place on the calendar, with the names serving as secondary titles accompanying their number.)
The Jerusalem Talmud5 tells us that the modern names of the months "came up [to Israel] with [the returnees] from Babylon," at the onset of the Second Jewish Commonwealth, approximately 350 BCE.6
So why did we begin to use these names? Why didn't we stick with the Biblical practice of referring to months by their number?
Nachmanides7 suggests that this is consistent with Jeremiah's prophecy: "Therefore, behold days are coming, says G‑d, and it shall no longer be said [by one who wishes to pronounce an oath], 'As G‑d lives, who brought up the children of Israel from the land of Egypt,' rather, 'As G‑d lives, who brought up the children of Israel from the northland [Babylon]...'"8
The original system was to count months in numeric order starting from the first month. Thus, any time a person mentioned a month he was in effect recalling the Exodus from Egypt—for now we are in, say, the sixth month—six months since the month of the Exodus.9 Thus the numeric naming served as a constant reminder of our deliverance from Egypt.
After we were delivered from Babylonian captivity, however, we started using the names that we came used to using in Babylon. And now, these names served to remind us that G‑d has redeemed us from this second exile.
I hope that I've been helpful today.
Yours truly,
Rabbi Menachem Posner