Why Babylonian Names for Jewish Months? asks this question. The basic answer given is that while the names were originally derived (in part) from Babylonian deities, they have become disconnected from those discarded idols and serve as a reminder of the Babylonian exile. When we lived in the exile, people began referring to the calendar dates by the names that were used by the people among whom they lived. After the end of the first exile, this continued because the major community continued to be in Bavel and those who settled back in Israel were still using it.
Additionally, it is suggested that it is a method of overturning the idol worship and converting it to "normal use" (see below abot Tamuz).
Consider the way we tend to use the current secular names in the world wide calendar nowadays even though they are based on the discarded myths of the Greco-Roman world.
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,’
but rather, ‘As G‑d lives, who brought up the children of Israel from
the north land [Babylon] . . .’”8
The original system was to count months in numeric order, starting
from Nissan. Thus, any time a person mentioned a month, he was in
effect recalling the exodus from Egypt: 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 became used to using in Babylon. And now,
these names served to remind us that G‑d has redeemed us from this
6. While many maintain that the names are actually taken from the
Babylonian tongue, the Rebbe maintains (Likkutei Sichot, vol. 23, pp.
214ff) that it is likely that many (if not all of) these names are
actually Hebrew, but that the practice of calling months by names
instead of their numeric position on the calendar originated in
Babylon. See also Tammuz—Time for Transformation.
7 Commentary to Exodus 12:2.
8 Jeremiah 16:14.
9 For similar reason, Nachmanides argues, we have no names for the days of the week. Sunday is called “the First Day,” Monday is “the
Second Day,” and so on—because we are constantly counting down to the
Shabbat. Every time we mention the day of the week—any day of the
week—we are fulfilling the divine precept (Exodus 20:8) to always
“remember the Shabbat day to keep it holy.”
Tammuz—Time for Transformation
Why would our sages allow the adoption of the name of idolatry into
the holiness of Judaism?
The short answer is that our role is not only to combat idolatry by
defacing it, because the psychological motivation that draws people to
idolatry is not cured that way. Instead, in the long run, we have to
transform the negative psychological proclivities that lead to
idolatry and transform them into positive ones. It seems therefore,
that the sages’ choice of the name of the false god Tammuz provides us
with a case study of the problem of idolatry and its solution.