I was thinking about this as it relates to the Braves. As a Braves fan of course I don't want to see that changed, but I also don't think the name itself is nearly as offensive as Redskins. So long as the Braves don't use the Screaming Indian logo I don't think it's nearly as offensive as the Chief Wahoo imagery either. To me (and I'm biased), the Braves are far less offensive than the Indians and Redskins by a fair margin. All that being said, I'm not sure I can objectively argue with people's distaste for the tomahawk chop, etc.
I even sat and thought about some new Braves names the other day and one thought I had was simply keeping the name, but removing the attachment to "indian culture." How you would do this and it not come across hokey, I don't know, but that's what I'd like to see if Atlanta is ever forced to drop their current mascot.