mid-13c., aquiten, "repay, reciprocate, reward or retaliate for" (a good or bad deed); c. 1300 as "satisfy a debt; redeem (a pledge)," from Old French aquiter, acquiter "pay, pay up, settle a claim" (12c., Modern French acquitter), from a- "to" (see ad-) + quite "free, clear," from Medieval Latin quitus, quittus, from Latin quietus "free" (in Medieval Latin "free from war, debts, etc."), also "calm, resting" (from PIE root *kweie- "to rest, be quiet"). Also in part from Medieval Latin acquitare.
From mid-14c. as "relieve (someone) of an obligation, release from a pledge," hence the meanings "set (an accused person) free from charges, pronounce not guilty," and "discharge one's duty; behave or conduct oneself" (for better or worse), all of which date to the late 14c. The notion in the word is "to release or discharge," from an obligation or from accusation, guilt, censure, or suspicion. Related: Acquitted; acquitting.