From
The Divine Code, Part II, topic 12:5 -
"It is permitted to mention the name of any idol that is written in the twenty-four Books of the Hebrew Scriptures..." (See
Sefer Ha'Chinuch, Commandment 86.)
The name Tammuz appears in the following verse:
"And He brought me to the entrance at the Gate of the House of the L-rd which was at the north; and there were there women sitting, bewailing the Tammuz" (Ezekiel 8:14)
In the context of this month in the middle of the summer, it is known as a month of tragedies which would lead to still greater tragedies, as commemorated by the Jewish fast day of the 17th of Tammuz. That day commemorates the breaching of the walls of Jerusalem by the Babylonians, and later by the Romans, which led up to the destruction of the First and Second Holy Temples, respectively. (It is also the day when the Golden Calf was worshiped and Moses broke the first Tablets of the 10 Commandments.)
Providentially, the Babylonian name for that month was Tammuz. The Jews adopted that name for the corresponding month in the Jewish calendar during the 70 years of the Babylonian exile. The appropriateness of the name should be clear. The Prophet Yechezkel (Ezekiel) was being shown by G-d the reasons for His great anger, namely, the various forms of foreign idol-worship which had been adopted by the Jews to replace the Divine Service in the First Holy Temple. This behavior was one of the main contributing factors in G-d's decree that the First Temple would be destroyed.
Thus, taking on this name for the historically tragic month became an eternal reminder to the Jews that they should never again let themselves be influenced to turn to idol worship, G-d forbid.