Argument
Middle English (in the sense ‘process of reasoning’): via Old French from Latin argumentum, from arguere ‘make clear, prove, accuse’.
wiktionary
From Middle English argument, [1] from Anglo-Norman and Old French arguement, from Latin argūmentum(“argument (for a position); evidence, proof; point, theme; thesis, topic; plot (in theatre)”), from arguere + -mentum( suffix indicating the instrument, medium, or result of something). [2]Arguere is the present active infinitive of arguō(“to argue, assert, declare; to make clear, prove, show; to accuse, charge with, reprove; to blame, censure; to denounce as false”), either ultimately from Proto-Indo-European *h₂erǵ-(“silver, white; glittering”), or from Hittite[script needed](arkuwā(i)-, “to make a plea, state one’s case”). The English word is analysable as argue + -ment.
The obsolete senses are derived from Middle English argumenten(“to argue, discuss; to consider, reflect”), [3] from Old French argumenter(“to argue”), from Latin argūmentārī, the present active infinitive of argūmentor(“to adduce arguments or proof, prove, reason; to adduce (something) as argument or proof; to conclude”), from argūmentum(“argument (for a position); evidence, proof; point, theme; thesis, topic; plot (in theatre)”) (see further at etymology 1) [4] + -or (the first-person singular present passive indicative of -ō( suffix forming regular first- conjugation verbs)).
The current sense is derived from the noun.
etymonline
argument (n.)
early 14c., "statements and reasoning in support of a proposition or causing belief in a doubtful matter," from Old French arguement "reasoning, opinion; accusation, charge" (13c.), from Latin argumentum "a logical argument; evidence, ground, support, proof," from arguere "make clear, make known, prove" (see argue). Sense passed through "subject of contention" (1590s) to "a quarrel" (by 1911), a sense formerly attached to argumentation.