Another answer is based on a Gemara in Masechta Shabbos 87b - the Gemara there discusses exactly when the tenth plague began.
Tosafos brings a Midrash from Shemos Rabbah and notes the following:
כשלקחו פסחיהם באותה שבת נתקבצו בכורות אומות העולם אצל ישראל ושאלום למה היו עושין כך אמרו להן זבח פסח לה' שיהרוג בכורי מצרים הלכו אצל אבותיהם ואל פרעה לבקש ממנו שישלחו ישראל ולא רצו ועשו בכורות מלחמה והרגו מהן הרבה הה"ד למכה מצרים בבכוריהם
When they took their Pesachim, the firstborns of the nations gathered to Yisrael and asked what they were doing. They said, "It is a Pesach sacrifice to Hashem. He will kill the firstborns of Egypt". [The firstborns] went to their fathers and to Pharaoh, to ask him to send out Yisrael, and they did not want to. The firstborn made a war, and killed many [Egyptians]. The verse "l'Makeh Mitzrayim bi'Vchoreihem" (He strikes Mitzrayim through their firstborn) alludes to this.
So from Tosafos we see that it is called Makkas Bechoros because it specifically alludes to a civil war that went on that centred around the firstborns indicated by the word Makkah. Indeed refer to Rashi on Tehillim 136:10