If it costs them more to keep the name than to change it, they will change it.
For the most part I agree, but I do think there's a wild card here...
Dan Snyder, while a businessman, is also a fan.
He's a person who grew up as a fan of this team from a very early age. He's also a very stubborn man from all accounts that I've heard and read about the man.
If somehow the Redskins name actually causes one of the most profitable sports franchises in the United States to STOP being profitable, then I agree...he'd likely take the initative to change it himself. But if it simply causes it to be LESS profitable, but still profitable, I can't see that causing Dan Snyder to change. The stubborn nature, mixed with his history as a FAN, I believe would cuase him to accept making less money but keeping the name.
Also, even if the team does make a lower profit then it currently is, changing the name doesn't inherently mean it'd fix it. A recent study by a university (I wish I could link to it, but I forgot the specifics and I originally heard about it on a radio show) tried to quantify what the most important motivating factors were for various sports franchises in the United States. Things like tradition, the product on the field, game day experiences, star players, etc.
Redskin fans were at the very top, along side Packers and Steeler fans, when it came to tradition being the primary motivating factor. And this shouldn't be surprising, as it's a fanbase still largely living on the joy of 20+ years ago.
Changing the name, unless it's accompanied with a SIGNIFICANT increase in product on the field over the next decade, could actually potentially cause a significant hit in terms of the profit of the team as the fanbase begins to feel a disconnect. It hampers the teams ability to prop up the past as a means of keeping interest in the present...something routinely done by the Redskins over the past 20 years.
IF a financial issue is going to cause a name change, it won't be a financial hit impacting the Redskins but rather the NFL. And I don't know if we're going to get to a place anytime soon where enough major sponsors will actually put pressure on the NFL, to a point where the NFL legitimately thinks it may be losing money from it, to cause a change.
If the name is going to change I largely feel it'll come about due to pressure from the Government in some substantial factor or pressure from the league due to financial issues....and the latter I don't see as being likely soon. I just don't see many avenues in which the Washington Redskins will suffer SIGNIFICANT financial hardships over the name.
It gets a lot of it's money via revenue sharing from the TV contracts. As it relates to individual sponsors, it's a big enough entity that I think even if it loses something like Fedex (and I don't think it will) I think it could recover well enough. It's money from ticket sales or merchandise is not likely to be largely impacted by the name thing, as fans of the team aren't largely stopping being fans because of the name. The product on the field is far more impactful in terms of tickets and merchandise then the fact they're not changing the name.