c. 1200, doten, "behave irrationally, do foolish things, be or become silly or deranged," also "be feeble-minded from age," probably from an unrecorded Old English word akin to Middle Low German and Middle Dutch doten "be foolish, be out of one's mind," all of which are of unknown origin, or directly from these words.
Century Dictionary and OED compare Dutch dutten "take a nap; mope;" Icelandic dotta "to nod, sleep;" Middle High German totzen "take a nap." Wedgwood writes, "The radical sense seems to be to nod the head, thence to become sleepy, to doze, to become confused in the understanding."
From late 15c. as "be infatuated, bestow excessive love." Also in Middle English "to decay, deteriorate," in reference to rotten timber, etc. (mid-15c.). There was a noun dote "fool, simpleton, senile man" (mid-12c.), but Middle English Dictionary considers this to be from the verb. Related: Doted; dotes; doting.