Where did "That's Mighty White of You" come from, and what does it mean?

In my experience, it has always been sarcastic. As Partridge notes, it means that one has all the “good” qualities of a white person. (Much as Huck Finn described Jim as “white inside.”) However, the whole notion that white people have inherent qualities or virtues superior to blacks (or other non-whites) is so silly that it is generally used as a parody of some stereotypical racist who would think that it actually was a compliment.

YMMV, I suppose.