c. 1300, "to make reasoned statements to prove or refute a proposition," from Old French arguer "maintain an opinion or view; harry, reproach, accuse, blame" (12c.), ultimately from Latin arguere "make clear, make known, prove, declare, demonstrate" (from a suffixed form of PIE root *arg- "to shine; white"). The transmission to French might be via arguere in a Medieval Latin sense of "to argue," or from Latin argutare "to prattle, prate," frequentative of arguere.
De Vaan says arguere is probably "a denominative verb 'to make bright, enlighten' to an adj. *argu- 'bright' as continued in argutus and outside Italic." He cites a closely similar formation in Hittite arkuuae- "to make a plea." The meaning "to oppose, dispute, contend in argument" is from late 14c. Related: Argued; arguing.