I don't care what the issue is — if your argument is that people are wrong for being offended, you don't have much fuel in the tank.
If you take offense to the offense, at least try to understand why someone feels that way. I consider myself very PC, but I've been accused of being insensitive at times. I've usually found those accusations unwarranted — and it's always shocking and hurtful, because not being that way is very important to me — but I've learned to check my ego and put myself in the shoes of the offended. Sometimes I reviewed what I said or wrote and thought, "Yeah, even though I meant this differently, I can see why someone would interpret it that way." Even if the person in question makes a point of being outraged by everything, I still take a look in case that isn't the whole story. Some allegations have merit and some don't. But I know who I am, and anytime I can learn to be better in expressing myself, that's a positive.
Yes, there are times when people are unreasonable with their offense. It's true that not every issue is tied to race, gender, sexual orientation, etc. But when they are, that's worth talking about. The Redskins name carries specific racial connotations; even if you believe that it's a tribute rather than a slur, you can't be surprised that some people find it offensive. It's an odd choice for anyone to think, "People are too sensitive about this."
Change the name.