Etymology
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aye (interj.)
word of assent to a question, 1570s, of unknown origin; perhaps a variant of I, meaning "I assent;" or an alteration of Middle English yai "yes" (see yea); or from aye (adv.) "always, ever."
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aye (adv.)

"always, ever," c. 1200, from Old Norse ei "ever" (cognate with Old English a "always, ever"), from Proto-Germanic *aiwi-, extended form of PIE root *aiw- "vital force, life; long life, eternity" (source also of Greek aiōn "age, eternity," Latin aevum "space of time").

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each 

Old English ælc (n., pron., adj.) "any, all, every, each (one)," short for a-gelic "ever alike," from a "ever" (see aye (adv.)) + gelic "alike" (see like (adj.)). From a common West Germanic expression *aina-galīk (source also of Dutch elk, Old Frisian ellik, Old High German iogilih, German jeglich "each, every"). Originally used as we now use every (which is a compound of each) or all; modern use is by influence of Latin quisque. Modern spelling appeared late 1500s. Also see ilk, such, which.

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either 

Old English ægðer, contraction of æghwæðer (pron., adv., conj.) "each of two, both," from a "always" (see aye (adv.)) + ge- collective prefix + hwæðer "which of two, whether" (see whether). Cognate with Old Frisian eider, Dutch ieder, Old High German eogiwedar, German jeder "either, each, every").

Modern sense of "one or the other of two" is late 13c. Adverbially, for emphasis, "in any case, at all," especially when expressing negation, by 1828. Use of either-or to suggest an unavoidable choice between alternatives (1931) in some cases reflects Danish enten-eller, title of an 1843 book by Kierkegaard.

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*aiw- 

also *ayu-, Proto-Indo-European root meaning "vital force, life; long life, eternity."

It forms all or part of: age; aught (n.1) "something; anything;" aye (adv.) "always, ever;" Ayurvedic; coetaneous; coeval; each; eon; eternal; eternity; ever; every; ewigkeit; hygiene; longevity; medieval; nay; never; no; primeval; sempiternal; tarnation; utopia

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit ayu- "life;" Avestan aiiu "age, life(time);" Greek aiōn "age, vital force; a period of existence, a lifetime, a generation; a long space of time," in plural, "eternity;" Latin aevum "space of time, eternity;" Gothic aiws "age, eternity," Old Norse ævi "lifetime," German ewig "everlasting," Old English a "ever, always."  

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hid (adj.)

early 13c., past tense and alternative past participle of hide (v.1).

How to entangle, trammel up and snare
Your soul in mine, and labyrinth you there
Like the hid scent in an unbudded rose?
Aye, a sweet kiss — you see your mighty woes.
[Keats, from "Lamia," 1820]
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chestnut (n.)

type of tall tree native to western Asia, southern Europe, and eastern U.S., also the large "nut" that it produces, 1560s, from chesten nut (1510s), with superfluous nut (n.) + Middle English chasteine, from Old French chastain (12c., Modern French châtaigne), from Latin castanea "chestnut, chestnut tree," from Greek kastaneia, which the Greeks explained as either "nut from Castanea" in Pontus, or "nut from Castana" in Thessaly, but probably both places are named for the trees, not the other way around, and the word is borrowed from a language of Asia Minor (compare Armenian kask "chestnut," kaskeni "chestnut tree"). In reference to the dark reddish-brown color, 1650s. Applied to the horse-chestnut by 1832.

Slang sense of "venerable joke or story" is from 1885, explained by U.S. actor Joseph Jefferson ("Lippincott's Monthly Magazine," January 1888) as probably abstracted from the 1816 melodrama "The Broken Sword" by William Dimond where an oft-repeated story involving a chestnut tree figures in an exchange between the characters "Captain Zavior" and "Pablo":

Zav. Let me see—aye! it is exactly six years since, that peace being restored to Spain, and my ship paid off, my kind brother offer'd me a snug hammock in the dwelling of my forefathers;—I mounted a mule at Barcelona, and trotted away for my native mountains. At the dawn of the fourth day's journey, I entered the wood of Collares, when, suddenly from the thick boughs of a cork-tree—
Pab. [Jumping up.] A chesnut, Captain, a chesnut!
Zav. Bah! you booby, I say, a cork.
Pab. And I swear, a chesnut—Captain! this is the twenty-seventh time I have heard you relate this story, and you invariably said, a chesnut, till now.
Zav. Did I? Well, a chesnut be it then. But, take your seat again.

Jefferson traced the connection through William Warren (1812-1888), "the veteran comedian of Boston" (and Jefferson's cousin) who often played Pablo in the melodrama.

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