To be foolishly, excessively fond, or to be feeble-minded with age, from Middle Dutch doten, to be silly or deranged.
Documented
The Dutch root
Documented: Middle English doten, from Middle Dutch doten or dotten, to be silly, to rave, related to Germanic words for folly and dizziness. Chaucer uses it; the double sense, foolish love and senile confusion, is original, because both are the same accusation: the mind gone soft. Dotage and dotard are its direct children.
Assessment: Documented: the Middle Dutch root and the Chaucerian usage are in the standard lexicons. The two meanings are one judgment.
Stories
The anti-dote joke
The word's entire modern fame: it is the presumed victim of antidote and the presumed identity of Christ in the resegmentation pun. A word that spent six centuries meaning senile affection became, overnight, a theology.
Assessment: Stories: the pun gives dote more traffic in a weekend than dotage gave it in a decade. The archive notes the comeback is ironic.
Interesting claims
The affection diagnosis
The old lexical insight the pun accidentally excavates: English once treated excessive fondness as a form of impairment, filed alongside madness. Dote and dote (senile) never split, the dictionary still holds them together, a fossil opinion that love past a certain dose is a derangement.
Assessment: Interesting: the fossil is still legible. The antidote pun is funny precisely because the word really does accuse its subject.
Sources and further reading
Oxford English Dictionary, dote: Middle English doten, from Middle Dutch doten, to be silly
Middle English Dictionary, doten: Chaucerian usage record
Bosworth-Toller and MED cross-references for dotage and dotard (the word family)
Worth asking
If doting is foolish fondness, is every grandparent technically afflicted?