The saying "Hair of the Dog"

Following on from Kosmo's answer, it also implies that, by drinking alcohol to cure your hangover, you're ingesting more of the same thing that hurt you in the first place.

The Wikipedia article for "Hair of the dog" cited in Kosmonaut's answer indicates that the notion that "like cures like" (as for example, the hair of a rabid dog cures rabies, or a shot of whiskey cures a hangover) is quite ancient, but the article no longer asserts that "The use of the phrase as a metaphor for a hangover treatment dates back to the time of William Shakespeare":

The expression originally referred to a method of treatment of a rabid dog bite by placing hair from the dog in the bite wound. Ebenezer Cobham Brewer writes in the Dictionary of Phrase and Fable (1898): "In Scotland it is a popular belief that a few hairs of the dog that bit you applied to the wound will prevent evil consequences. Applied to drinks, it means, if overnight you have indulged too freely, take a glass of the same wine within 24 hours to soothe the nerves. 'If this dog do you bite, soon as out of your bed, take a hair of the tail the next day.'" ...

I don't know why the sentence about "the time of Shakespeare" was excised, but relevant instances do occur in English at least as early as 1545. From John Heywood, A Dialogue Conteinyng the Nomber in Effect of All the Prouerbes in the Englishe Tongue Compacte in a Matter Concernyng Two Maner of Mariages (1546):

Early we rose, in haste to get awaie. / And to the hostler this mornyng by daie / This felow calde, what how felow, thou knaue, / I praie the leat me and my felowe haue / A heare of the dog that bote vs last nyght. / And bytten were we both to the brayne aryght, / We sawe eche other drunke in the good ale glas, / And so dyd eche one eche other, that there was.

Other early variants on this expression (courtesy of Morris Tilley, A Dictionary of the Proverbs in England in the Sixteenth and Seventeenth Centuries, 1950) appear in Randle Cotgrave, A French and English Dictionary (1611/1673):

Prendre du poil de la beste. To take a remedy for a mischief from that which was the cause thereof; as to go thin cloathed when a cold is taken; or in drunkenness to fall a quaffing, thereby to recover health, or sobriety, near unto which sense our Ale-knights often use this phrase, and say, give us haire of the dog that last bit us.

And in Ben Jonson, Bartholmew Fayre (1614/1631):

Qvarlovs. No, and I had, al the Lime-hounds o'the City should haue drawne after you, by the sent rather, Mr. John Little-wit! God saue you, Sir. 'Twas a hot night with some of vs last night, Iohn: shal we pluck a hayre o' the same Wolfe, to day, Proctor Iohn?

And in "Sat{ira} 5 {Against Gluttony, Drunkenness, and Tobacco}" in Times' Whistle: Or, A Newe Daunce of Seven Satires, and Other Poems (1616):

After some 3 howers sleepes strong operation / Hath brought their braines into a better fashion, / They gin to wake, & finding themselves ill / Of their latest surfet, which hath force to kill / The strongest body, to 't afresh they goe, / To drink away their paine; such heartsick woe / By an immoderate drunkennesse procurde, Must by "a haire of the same dog" be curde.

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The expression in early proverb collections

David Fergusson's 1598/1641 collection of Scottish proverbs evidently includes this line, although I can't find a viewable copy of the text online:

Tak ane hair of the dog quhilk bait yow yisternight;

A subsequent English proverb collection, J.H., Proverbs, or Old Sayed-Sawes and Adages in the English Toung (1660) makes clear that the expression didn't mean simply to take a single drink to palliate the symptoms of a hangover:

to take a hair of the same dogg; viz. to be drunk with the same drink again.

And John Ray, A Collection of English Proverbs (1678) has much the same entry:

To take a hair of the same dog.

i.e. To be drunk again the next day.

Mention of "hair of the dog" also appears in James Kelly, Complete Collection of Scottish Proverbs (1721), this time with the curative aspect of the treatment suggested:

Take a Hair of the 'Dog that bit you.

It is suppos'd that the Hair of a Dog will cure the Bite he gives. Spoken to them who are sick after Drink, as if another drink would cure their Indisposition.

In light of this lengthy and widespread usage, it is interesting that John Jamieson, Supplement to the Etymological Dictionary of the Scottish Language, volume 1 (1825) seems to consider it something of a rare bird. Jamieson's discussion of the phrase is worth quoting at some length:

A hair of the Dog that bit one, a proverbial phrase, metaph[orically] applied to those who have been intoxicated, S. [Quotation of the entry for the expression in James Kelly's proverb collection omitted.]

This phrase is not unknown in England; although I have met with no example of the use of it except in the Dictionaries of Cotgrave, Ludwig, and Serenius. They all give the same sense with that above mentioned. ...

So absurd did this phrase seem, that I would never have thought of investigating it, had I not accidentally met with a passage in a publication, the writer of which could have no end to serve by relating what was totally unfounded, and so unlike the apparent simplicity of the rest of the narrative.

Having mentioned that, when at Wampoa in China, his dog Neptune had bit a boy, who was meddling rather freely with the articles belonging to him, and that he "dressed the boy's hurt, which was not severe," he adds; "In a short time after I saw him coming back, and his father leading him. I looked for squalls, but the father only asked a few hairs out from under Neptune's fore leg, close to the body; he would take them from no other part, and stuck them all over the wound. He went away content. I have often heard, when a person had been tipsy the night before, people telling him to take a hair of the dog that bit him, but never saw it in the literal sense before." J. Nicol's Life and Adventures, Edin[burgh] 1822, p. 100.

Jamieson thus seems to have believed that the the phrase in his day was in more widespread usage in Scotland than in England.

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The expression in medical texts

One early instance of "hair of the dog" in the English English Books Online database seems to address to the traditional protection against rabies. From T. K., Doctor in physick, The Kitchin-Physician, or, A Guide for Good-Housewives in Maintaining Their Families in Health ... (1680). Unfortunately, I don't have access to the full content of this book, but the table of contents does include a chapter titled "Against the Biting of a Mad-Dog" and doesn't include one dedicated to excess of drink.

But the phrase again appears in the context of a hangover treatment in Edward Eizat, Apollo Mathematicus, or, The Art of Curing Diseases by the Mathematicks According to the Principles of Dr. Pitcairn (1695), where the author rails sarcastically against "like cures like" prescriptions:

One would think, that while the Doctor is speaking of sleepy Diseases, and the Qualities of Opium, that he were asleep himself, or had taken a Grain or two of the Juice of the Poppie, this Discourse looks so like Raving. For if the Cause of Sleep, and sleepy Diseases, be the Rarifaction of the Blood, then every thing that rarifies the Mass of the Blood would occasion these Diseases, that is, every thing that is proper for Curing the Apoplexy, &c. would likwise occasion it: As Spirit of Hart-horn, salt Armoniack, Oyl of Amber, Rosemary, Tincture of C•ftot, &c. which are Remedies used with good success in those Diseases, and rarify exceedingly the Mass of the Blood, would inevitably either bring on or increase the Malady: this is one of the most wonderful and useful Discoveries that ever the World was blest with, and will make any Body a Physician in a trice: for when you come to a Patient, you have no more to do, but ask him what was the occasion of his Sickness? Was it a surfiet of a Venison pastie? then be sure to eat as much the next day, and you shall be Fish-whole. Was it a surfiet of Drink? then take a Hair of the Dog that bit you: for

Si tibi nocuerit hesterna potatio vini

Cras iterum bibes &; erit tibi medicina.

If Sickness caused by Drink you mind to Cure,

To drink just as much the next day be sure.

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Conclusion

The notion that "like cures like" goes back to ancient times, and the application of this general precept to dog bites and dog hair seems to be very old as well.

The metaphorical application of "hair of the dog" (or more precisely "a heare of the dog that bote vs last night") to the notion of dealing with the aftereffects of excessive drinking by engaging in renewed excessive drinking goes back in English at least to John Heywood's Dialogue Conteinyng the Nomber in Effect of All the Proverbes in the English Tongue Compact in a Matter Concernyng Two Maner of Mariages (1546). Whether it began in England and moved from there to Scotland or vice versa is unclear, but at least one authority, writing in 1825, seems to think that the expression was far more prevalent at that time in Scottish usage than in English usage.

The idiomatic phrase has persisted, I suppose, because the idea of curing a hangover by drinking more alcohol possesses perennial appeal to generations of drinkers. And an evergreen (if not everclear) idea, strikingly expressed, is the stuff that proverbs and idioms are made of.