H. P. Grice, "The etymology of an argument"

arguere "make clear, make known, prove, declare, demonstrate," from PIE *argu-yo-, suffixed form of 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."