mid-14c., amonesten "remind, urge, exhort, warn, give warning," from Old French amonester "urge, encourage, warn" (12c.), from Vulgar Latin *admonestare, from Latin admonere "bring to mind, remind (of a debt);" also "warn, advise, urge," from ad "to," here probably with frequentative force (see ad-) + monere "to admonish, warn, advice," from PIE *moneyo-, suffixed (causative) form of root *men- (1) "to think."
The -d- was restored on Latin model in English as in French (Modern French admonester). The ending was influenced by words in -ish (such as astonish, abolish). Related: Admonished; admonishing. Latin also had commonere "to remind," promonere "to warn openly," submonere "to advise privately."