A substance that counteracts poison, from Greek antidoton, something given against: anti, against, plus didonai, to give.
Documented
Given against
The etymology is clean and documented: Greek antidoton, neuter of antidotos, given against, from anti plus didonai, to give. Latin absorbed it as antidotum; French and English followed by the fifteenth century. The word entered English already fused, a dose given against a poison. There was never an English dote for it to oppose.
Assessment: Documented: Liddell-Scott's Greek lexicon, the OED, and every Romance cognate agree. The word arrived whole.
Stories
The anti-dote split
The folk resegmentation: anti-dote must mean against dote, so dote must be a poison, a folly, or, in the comment-section version, Christ. Dote is a real word, meaning to be foolishly fond, from Middle Dutch doten, to be silly, but it shares no root with antidote. The joke is a false-friend accident between a Greek compound and a Dutch verb.
Assessment: Stories: the pun is traceable as a genre, soundalike resegmentation, not as doctrine. Nobody has ever been cured of doting.
Interesting claims
The poison-cure mirror
The deeper pattern: antidote belongs to a family of words where the cure names itself against the disease, antibiotic, antiphon, antagonist. The pun exposes how easily speakers re-carve old words at the seams that sound modern, which is exactly how real folk etymologies are born.
Assessment: Interesting: the joke is a live demonstration of the mechanism the archive exists to document.
Sources and further reading
Liddell-Scott-Jones Greek Lexicon, antidotos: given against, anti plus didonai (primary lexical source)
Oxford English Dictionary, antidote: from Latin antidotum, Greek antidoton; English from the fifteenth century
Merriam-Webster, antidote etymology (the given-against compound)
Centre National de Ressources Textuelles et Lexicales, antidote (the Romance transmission)
Worth asking
If the cure is anti-dote, what was the disease of the word doing before the prefix arrived?