Early uses of the phrase “to a fare-thee-well” and its earlier variant “to a fare-you-well” found in newspaper archives provide a possible theory: that the expression first meant besting someone by a lot, as in until they have to say farewell, and from there drifted to simply mean doing something a lot or to the extreme.
Early uses of “fare-thee-well” in this sense are sports related. The earliest I found is from 1899 and can be understood as an extension of the literal meaning, “farewell.”
Hughey, one of the leading twirlers of the Browns in ’98, pitched for the “Exiles” and was batted to a “fare thee well.” The St. Louis batsmen made a total of 20 hits of his delivery.
If variations of “X was [Verb]ed to a ‘fare-thee-well,'” where “X” is defeated or overwhelmed severely, were the form in which the phrase grew into popularity, it would provide an explanation for how it came to mean “to the max,” as batting or pitching someone “to a fare-thee-well” presumably means they were beaten “hugely.”
The earliest “fare-you-well” in the same sense that I found was only slightly older, from 1893, and seems to lend support for the notion that the phrase originally meant besting someone by a large degree, this time in gambling / horse-racing.
For some time there has been talk of a race with “Ben Harrison” against the so-called world beater, “Lizzie Norton,” and all on account of the backers of the mare making a $300 crack that she could beat “Ben” to the half. This bluff was called to a fare-you-well, and word was sent back they would take the bet and any more they might want to come in with.
In this case, “to a fare-you-well” would probably not work if readers were not already familiar with the expression, as it does not carry quite the literal meaning the baseball citation does. Still, these early citations that imply besting someone to the extreme [such that they figuratively or literally are forced to leave] appear similar to the sense “doing something to the extreme,” and could possibly explain the origin of the sense.
A farewell is also an expression of good wishes at a parting. If you’re leaving a job after being there a long time, your co-workers might throw you a farewell party. Farewell is an expression, like “goodbye,” but also a noun — if you like to leave without a lot of fuss, you don’t like long farewells. Farewell is a combination of fare and well. Fare comes from the Old English word faran, which means “to journey.”