B”H
Mordechai מרדכי
Root Meaning #1: From the Aramaic words מירא דכיא which translate as Pure Myrrh which was a pre-eminent spice that was used in the Temple as part of the anointing oil (Source: Talmud Chullin 139b)
Root Meaning #2: There is some speculation that this name was a derivative of the Babylonian idol Marduk (and Mordechai had another Hebrew name either Pesachya or Malachi) Source: Steinsaltz Megillah
Let’s unpack the Root #1:
Mordechai is referred to in the Megillah as Mordechai HaYehudi, which means Mordechai the Judean, or Mordechai the Jew. While we are familiar with the term Jew (again a shortened form of the word Judean), a Judean technically would refer to a Jewish person specifically descending from Judah, one of the twelve tribes. However, Mordechai descended from the tribe of Benjamin, not Judah,(the Megillah also mentions he had Benjamin lineage) so why does he have that title?
The simple answer In the 5th century BCE, the Kingdom of Israel was conquered by Assyrian King Sennaherib, and the vast majority of the ten tribes were exiled and lost. The bulk of the remaining Israelites were the residents of the Kingdom of Judah, and the term "Yehudi" or "Jew" came to refer to all the Israelites, regardless of their tribal ancestry.
But there is also a deeper meaning to the name "Judean/Jew" and this will also shed light upon the life mission of Mordechai. The Talmud (Tractate Megillah 12b) asks: "Mordechai is called a Yehudi, implying that he descended from Judah; he then is called Yemini, implying that he is a Benjaminite!" Rabbi Jochanan responds: "He was a Benjaminite. Yet he was called a Yehudi because he rejected idolatry—and anyone who rejects idolatry is called a Yehudi."
The commentaries explain that the name Yehudah shares the same root as the Hebrew word hoda'ah, which means acknowledgment or submission to something greater than oneself (i.e. G-d). One who acknowledges G‑d's existence and submits to His authority—to the extent that he is willing to sacrifice his life for the sanctification of His name—he is called a Yehudi.
So now we know why Mordechai is called a Yehudi/Judean/Jew which derives from submission to a Higher authority and purpose. He exemplified the deep inner connection a Jew feels for G-d and the Jewish people, even willing the lay his life on the line for both. As a matter a fact, he even brought out that feeling of submission to a Higher power in the rest of the Jews of his time (of which many were highly assimilated).
But what does that have to do with his name Mordechai, which we see is connected to the preeminent spice Pure Myrrh that was used in the anointing oil in the temple?
It is interesting to note that Pure Myrrh was from a secretion from the blood of a male musk deer which is not a kosher animal. How was a non-kosher ingredient used for one of the holiest purposes possible? The answer is that the processing of the blood of the male musk deer transformed it to be kosher.
Spiritually, this represents the transformation of our animal drive/ego to become a tool for our Divine mission. How is that possible? How can the ego become a tool for holiness?
The answer lies in the name Yehudi: thought submitting ourselves with utter devotion and humility to our Higher Power, even the intial unsavory parts of us get transformed into a holy purpose. Source: Torah Ohr 99a-b
Sources: The Kehot Megillah Page 14b