That's mighty white of you...."

I've looked around and there are many people who think it's racist because of the impression that it means to be act like a good person, and the association between being a good person and being white is seen as racist. As Hot Licks mentioned one of the most common meanings going around is that the person thinks they did something altruistic or helpful, when in fact their action wasn't appreciated much or at all. For this reason it can have a connotation of thinking you did something good but are actually oblivious to the fact that you were useless. So it's possible it doesn't have racist meaning at all.

I found this meaning in Urban Dictionary and the message board of phrase.org.uk:
phrase.org.uk
Urban Dictionary

However the Wikipedia entry doesn't mention this particular negative association with the term:

A similar American expression is That's mighty white of you, with the meaning of "thank you for being fair".
Play the white man

Furthermore on the phrase.org.uk page someone gave an excerpt from Eric Partridge's "Dictionary of Catch Phrases American and British, and it didn't seem to have the connotation of being unhelpful, but possibly a racist one:

Of the US usage, Prof. John W. Clark, 1977, has noted that it was, at first, used seriously--'like a white man, not like a Negro., it just seemed to mean a good or generous person.
phrases.org.uk

Sorry I couldn't find the primary source for that dictionary.

So I can't find a definitive meaning for this one.

Also I found this: Clint Eastwood (Dirty Harry) saying "That's mighty white of you" directed I think to a black person. I don't know the context of this one, maybe you can investigate it or figure it out.

That's mighty white of you

I just remembered we had a brand of bread called Mighty White. I'm unsure if it was an allusion to this. These ads are from the late 80s, but I'm pretty sure they were sold into the mid 90s, I think.

https://www.youtube.com/watch?v=g1T8_M8Fjy0
https://youtu.be/WRIDroRbDHc?t=2m12s

Their website is still up, no idea if they still sell it. It's marked copyright 2015 and isn't working 100%, so it may be no longer. In today's weird PC world I wouldn't be surprised if people took offence at the mere name of this bread. All I know is they don't sell it where I'm from anymore.

In comments, Hot Licks wrote:

It's not expressing racial superiority -- quite the opposite. It implies that the target of the epithet is behaving as if he were "entitled" to something. Ie, he is behaving as if he were better than everyone else (when, in fact, he's quite clearly an idiot).

It's a relatively rare term, and likely limited to black and middle-class white speakers (and middle-class whites are rarely "notable" in literature). I first heard it in Minnesota ca 1975 (it would have been considered "radical" in Kentucky where I was born), and have only heard it maybe a half-dozen times since (though I've probably used it myself a few dozen times).

mighty white of you OED

c. colloq. (orig. U.S.). [Initially] Honourable; square-dealing. Also as adv. Now somewhat rare and generally regarded as offensive.

As in:

2004 T. C. Boyle Inner Circle i. i. 26 I never really got to thank you for what you've done..—it was really white of you.

As recently as 2004 (as in the above) the sense was 'fair'. Now it is rarely used as the working is considered offensive.

Originally used under colonialism and before civil rights, this phrase expressed appreciation for honorable or gracious behavior, under the assumption that white people were inherently more virtuous. Today, it is generally used sarcastically in reference to underwhelming acts of generosity.

-urban Dictionary, which conforms to how I’ve always heard it used

The OED lists the meaning as sense 5c of the adjective white. The OED says it is of American origin. Indeed, as a British person born during WW2, I don't recall having ever heard it used - though I have certainly heard and have been familiar with the term white man used similarly e.g. "he acted the white man and did the decent thing". The OED also has a separate entry for this quoted below.

The notion of white being good, clean and innocent, and black being diabolical, and bad is deeply ingrained in Anglo (and no doubt other European) cultures. Metaphors and expressions such as "pure as the driven snow" are manifold in English.

White

5c. colloq. (orig. U.S.). Honourable; square-dealing. Also as adv. Now somewhat rare and generally regarded as offensive. Cf. white man n. 3. In origin probably chiefly reflecting racial and cultural stereotypes formerly associated with European descent (and hence implying contrast with people of other races), although perhaps partly informed also by sense A. 7a.

1837 M. Huxley in T. M. Cooley Sketches Life & Char. L. Haynes iv. 73 ‘The preacher had not proceeded far in his sermon,’ said the man, ‘before I thought him the whitest man I ever saw.’

1865 ‘M. Twain’ Sketches (1875) 74 The parson..was one among the whitest men I ever see.

1876 W. Besant & J. Rice Golden Butterfly II. v. 83 A good fellow is Rayner; as white a man as I ever knew.

1913 E. Wharton Custom of Country xviii I meant to act white by you.

1948 K. S. Prichard Golden Miles 374 Tom Gough's one of the finest, whitest men ever drew breath. There's not two like him born in a century.

2004 T. C. Boyle Inner Circle i. i. 26 I never really got to thank you for what you've done..—it was really white of you.

White man

3. slang (orig. U.S.). A man of honourable character. Now somewhat rare and generally regarded as offensive. Cf. white adj. 5c. 1883
   Cent. Mag. 26 913/1 You've behaved to me like a white man from the start. 1887 Pall Mall Gaz. 22 June 5 Tricoupis the President is a white man—an extremely white man. 1936 ‘F. Gerald’ Millionaire in Memories iv. 114 I shouldn't have stayed as long as I did if I hadn't met two ‘white men’—the definition of a ‘white man’ in this case being ‘a decent soul’.

The term used to refer to a deed done with pure intentions. White being the colour of purity. Still used for christenings, wedding dresses, good vs evil in characteristics Society turned it racist