What is the meaning of "Fare thee well"?

Solution 1:

"Fare thee well" is an expression of good luck used at a parting. So the meaning is similar to "good luck". It contains the archaic second-person pronoun "thee". It's not a common expression: it's a little more common to say "farewell", although that still sounds very formal.

The Oxford English Dictionary entry for the verb "fare" categorizes the grammar of this construction as an impersonal verb followed by a dative pronoun. A "dative" pronoun is one with the sense of "to X", so "Fare thee well" apparently would be more or less equivalent to "May it fare well to thee". That is explained as the meaning of the expression in Complete manual of parsing, by W. Davidson and J.C. Alcock (1875, p. 159).

The archaic verb "fare" has a somewhat similar meaning to the verb "go" so it could be seen as an old-fashioned way of saying "I hope things go well for you."

Solution 2:

There should be no a real problem with this phrase.

Fare meant in earlier times "to make one's way", "to journey".

Thee is the second person object pronoun (now usually "you").

Well might be thought of as adverbially "good".

Fare thee well means to "live well", "have a good life". Some still used the phrase when I was young. It was used as a valediction when people parted for an extended period. It has been misunderstood because of the idiomatic phrase "to a fare thee well", which came to mean "done perfectly". I think it is clear that meaning came from the idea something was done to a point there was nothing left but to say "goodbye". (Sam beat Fred "to a fare thee well", (perfectly beat him), meaning there was nothing left to do but say goodbye.)

Solution 3:

By itself, "fare thee well" a simple phrase of parting, a variant of "farewell" or "farewell [to] thee", and that's what it means in the song lyric. As a phrase "to a fare-thee-well" it means "to an extreme degree".