Q:

"I have X in my blood" arguments

Lately, I have been hearing the argument "Oh, well I have Indian(Native American) in my blood, and 'redskin' does not offend me so why change the name?" coming from people who claim their great-great-great grandmother was from one of the hundreds of tribes from the area. 
I know it's been an argument made before, and I've always found it to be silly. There are some obvious arguments to be made from that statement: They haven't experienced life the same way as someone who is 100% native american, or that one person from a large group is not a representative of said group. 
But now that I am hearing this claim being made more and more, I was wondering what fallacy this would fall under? It seems to be similar to this argument, "I have X friends so I can't be racist against X people." But I feel it is a bit more than that? 

A:

This can be a case of the Cherry Picking fallacy:

Cherry picking, suppressing evidence, or the fallacy of incomplete evidence is the act of pointing to individual cases or data that seem to confirm a particular position, while ignoring a significant portion of related cases or data that may contradict that position. 

The person making the argument picks a case which supports his point, ignoring the cases that don't support his point.

A:

In my experience, this is not a logical error, so much as an unarticulated moral premise.
The speaker holds an underlying assumption here that if it is possible not to be bothered by this kind of thing, one is morally obligated not be bothered by it, or at least to do one's best to not act upon being bothered.
Using himself as evidence this is possible, he is ashamed of the offended members of his group, who are not trying hard enough to get along with the rest of the world.
He is presuming the ultimate value of some version of democratic social harmony: non-interference in the majority's autonomy, 'Give the benefit of the doubt' or 'Go along, get along'.  And this is in the foreground to a degree that he denies consideration to more important moral concerns.
You can tell the difference between this and suppression of evidence or 'cherry picking' because, if you render the argument statistical, it does not matter unless the numbers are hugely predominant, and often even then.
He is not unaware he is an exception, he embraces the exceptionality and considers it morally superior.

