The word for a political killer began as a medieval insult. The drug story attached to it is the part almost everyone gets wrong.
The Nizari Ismailis were a branch of Shia Islam. In 1090 their leader, Hasan-i Sabbah, took the fortress of Alamut in the mountains of northern Persia. The Nizaris later held castles in Syria too, where the Crusaders met them. Their Syrian chief Rashid ad-Din Sinan became known in Europe as the "Old Man of the Mountain." The sect was feared for targeted killings of powerful enemies.
Their opponents had a name for them: hashishi, plural hashishiyya or hashishin. Wikipedia traces the earliest recorded use to 1122, when the Fatimid caliph al-Amir used it against the Syrian Nizaris. On its face it means "hashish user." According to the Institute of Ismaili Studies, it was a term of abuse meaning something like "low-class rabble." It said nothing about drugs.
The Crusaders picked the word up in Syria and made it the sect's name. They also took it literally. Tales spread that the killers were drugged before their missions. Marco Polo added a secret "garden of paradise" where recruits were supposedly given a taste of heaven. The Institute of Ismaili Studies notes that hostile Muslim writers of the time, who had every reason to repeat such a charge, say nothing about it. Etymonline puts it plainly: there is no evidence that the medieval Ismailis used hashish.
The spelling holds a second mistake. The Arabic plural ending -in was taken by Europeans as part of the word itself. That is why English has assassin rather than something closer to hashishi.
In 14th-century Italy the word broadened from the sect to any murderer of a public figure. It moved into French and reached English in the 1530s.
The hashish link got a scholarly stamp in the 19th century. The French orientalist Silvestre de Sacy connected the word to the drug, and the explanation stuck in European books for generations. Modern historians of the Ismailis, such as Farhad Daftary, have spent decades taking those legends apart.
So the etymology is sound: assassin really does come from the Arabic for "hashish user." It is the story behind the name that is the myth.