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

Solution 1:

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

Solution 2:

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.