Former season ticket holder here (keep the jokes to yourself please).
Like many things that one grows up with, you do not give it a second thought. I was 5 and watched football with my father; his team was the Redskins and so was mine. I never gave it a thought that the term meant anything other than the name of the football team. Football = Redskins to me and to many Washingtonians. Similarly, the name of a street in Northern Virginia (Lee Highway named for Robert E. Lee) was always just a street in NoVa. Heard it on the traffic reports every day - a crash on Lee Hwy, what else is new. I do not think that any malicious thought was given when the Redskins or Lee Hwy were named.
Even if Redskins was a slur at one time, maybe it is not now - can't the meaning of words evolve. When was the last time you heard someone (other than a football player for Washington) called a Redskin, and done so in a negative manner? As an example I offer a word whose meaning has changed. That word is niggardly. Originally meaning stingy or miserly, its origin is from the Middle Ages. Over time, a derivation of it was used in a derogatory (to say the least) sense since the mid 1500's and really picked up steam in the 1700's. In modern times, one cannot use the word niggardly without it being confused with it's derivation. In fact there are a few incidences where niggardly was used correctly in public but was misunderstood to mean something else and the user faced serious repercussions. Its use by a Washington DC aide to Mayor Williams in the late 1990's ended up getting the aide fired (though was offered his job back) for using that word in reference to official DC business. Interestingly enough, Julian Bond (then chairman of the NAACP) was taken aback by the backlash of the use of that word and reportedly said "You hate to think you have to censor your language to meet other people's lack of understanding".
Up-thread, someone mentioned "how many need to be offended to change the name". That is a great question. How many is the right amount, and how does one determine that amount? I'd venture to guess you can get 10% of any group to object to something. Further, you might be able to drill into most any team name to find something objectionable - where is the line drawn? American history is littered with great stories of Cowboys - but who was involved to a large degree with the destruction of the Native Americans? Rangers is a nice name for a team, yet there are many stories of Texas Rangers lynching Latinos, some as recent as the early 1900's. Should the Rangers look for a new name? The Redskins are tops on most everyone's list for team names to change. So it looks like it will eventually get changed - then what is next: Indians / Chiefs / Warriors / Braves - get rid of all Native American names and imagery. OK, fine. Then what is next after that - where is the line? Cowboys, Rangers, Canucks, Fighting Irish - anyone else?
So what is my stance on the name change? To be honest, I am indifferent. Change it, don't change it, whatever happens will happen. If they are changing - go to some sort of an animal - anythign else will just kick the can of "change the name" a little further down the road.