Customarily, the expression “a hair of the dog that bit you” applies to a drink taken as a pick me up on the morning after a spree, to a drink taken for relief from an excess of drinks.
Heywood, in 1546, thus recorded it in his “Dialogue conteynyng prouerbes and epigrammes”: “I pray the leat me and my felow haue A heare of the dog that bote vs last night, And bitten were we both to the braine aright.”
The curious name for the practice comes from a widely accepted medical doctrine that goes back at least to the sixteenth century and was probably the common folk belief many centuries before that.
That is, it was generally and seriously believed that if one were bitten by a dog suffering from rabies (by a “mad dog”), one’s chance of recovery was greatly improved if a hair from that dog could be secured and bound upon the wound.
It may be pertinent to remark that, though this treatment was still recommended up to the middle of the eighteenth century, its efficacy is now doubted; possibly the same could be said of the morning pick-me-up.