This question is addressed using the same comment of Tosafos that is @JoelK's jumping-off-point, in Y'sodos Hat'fila (p. 181). Among the other answers, the most compelling one is that between minchas haboker and minchas ha'erev, the latter is more strongly associated with the name "mincha".
The showdown between Eliyahu and the Ba'al adherents demonstrates this association (18:26-36). They were active "מֵהַבֹּקֶר וְעַד הַצָּהֳרַיִם" - "from morning till afternoon", after which Eliyahu responded confrontationally "בַצָּהֳרַיִם" - "in the afternoon". The next time period mentioned is bounded by "כַּעֲבֹר הַצָּהֳרַיִם" - "[the end] of the afternoon" - at its start and "עֲלוֹת הַמִּנְחָה" - "the mincha offering" - at its end, meaning that the unqualified word "mincha" must be that of the evening.
Later (16:15), when the navi records the offerings of Achaz, it says
וַיְצַוֶּה הַמֶּלֶךְ־אָחָז אֶת־אוּרִיָּה הַכֹּהֵן לֵאמֹר עַל הַמִּזְבֵּחַ הַגָּדוֹל הַקְטֵר אֶת־עֹלַת־הַבֹּקֶר וְאֶת־מִנְחַת הָעֶרֶב וְאֶת־עֹלַת הַמֶּלֶךְ וְאֶת־מִנְחָתוֹ וְאֵת עֹלַת כָּל־עַם הָאָרֶץ וּמִנְחָתָם וְנִסְכֵּיהֶם וְכָל־דַּם עֹלָה וְכָל־דַּם־זֶבַח עָלָיו תִּזְרֹק וּמִזְבַּח הַנְּחֹשֶׁת יִהְיֶה־לִּי לְבַקֵּר׃
, in which the words ola and mincha are used repeatedly. When the two appear in parallel to refer to the morning and evening offerings (emphasis mine), only the evening one is referred to by the term "mincha". Subsequently the word presumably has the more common meaning (among those referring to specific normative offerings and not generic ones or gifts) of attendant flour offering.
Indeed, the only two times the phrase "michas haboker" appears it refers to the attendant flour offering that was brought along with the morning ola and not the karban itself. (I might be misinterpreting or overinterpreting this point.)
On the other hand minchas ha'erev is used several times in addition to the Eliyahu one to refer to the main evening offering (e.g.).
Along similar empirical lines, Rav Ya'akov Emden doubles down (bolded below) on the original answer of Tosafos and seems to share some of @Joel K's hesitancy about it, based on the disclaimer with which he concludes the discussion: (Lechem Shamayim at the beginning of the 4th perek of B'rachos, pp. 43-45)
שלשחרית יש שם אחר ולא נמצא במנחה שם זולתו. מלבד שיתכן שיש לרז"ל עוד בעצם המלה כוונה נעלמת
...that shacharis has another name, but there is no name for mincha other than that. Plus the possibility that our sages had a secret intention with this particular word.
(He also mentions and rejects as far-fetched Ramban's suggestion that "minchas haboker" in Tana"ch means mincha g'dola while "minchas ha'erev" means mincha k'tana so the word exclusively refers to times in the afternoon or evening.)
In a way this just pushes the question back to "why did the convention settle on that mincha as the generic one?" but at least it shows it as a phenomenon in Tanach and not just based on it.