What's in a name? A lot, is the answer, if you're a famous American professionals sports franchise called the Redskins. I refer, of course, to my local National Football League team, the Redskins. For many, maybe most, Washingtonians, they're the most cherished institution in town. But they're also, according to detractors, a billboard for a racial slur.

We've been round this course before, but this time the din is especially loud. The reasons are various. For one thing, the team is doing better, so more people notice it. Pressure, too, has been growing for the Redskins to move back into DC proper from suburban Maryland. But, if so, Washington's mayor has suggested, then a name change should be considered. And finally, there was this month's symposium at the National Museum of the American Indian here, when speaker after speaker took issue with Native American mascots in professional sports, and the Redskins in particular.

It's easy to dismiss the fuss as political correctness in America once again run riot. The name, it is pointed out, has been around since even before the Redskins moved from Boston to Washington in 1937. Is it really offensive, and even if it is, what's the problem? Yes, American Indian activists wax indignant about it – but the most recent poll on the subject, admittedly a decade ago, found the average Native American basically couldn't care less.

Nor, too, could the Redskins' management, adamantly opposed to change. As is currently noted on the franchise's website, some 70 high school teams, from 25 US states, go by the same name. Far from being a slur, it is intended as a tribute to the valorous warrior qualities of the original Indians – the virtues on which a fierce sport prides itself most.

Still, the critics have a solid case. Could you imagine that museum where the symposium took place, in its hallowed place by the Washington Mall, being called the National Museum of the Redskin? Or try a few comparable variants (the precise degree of offensiveness is naturally subjective): the Washington Darkies, say, or the Washington Eyeties, the Washington Ragheads, the Washington Spics. No way.

Dan Snyder, a media tycoon who owns the Redskins, is Jewish. As The Washington Post columnist Mike Wise told the symposium, Snyder surely would not appreciate it if his team were named the Hebrews, and its mascot was a man "with the Dead Sea Scrolls in one hand and a menorah in the other".

And as for the "honour" supposedly being bestowed – is that not just a condescending trifle tossed by the conqueror in the direction of the noble savage, scant recompense for the appalling treatment of indigenous Americans in the 19th century by the European settlers and pioneers?

Slowly but surely, the revisionists seem to be winning the battle. Not perhaps at the major league level, where Chief Wahoo is still the public face of the Cleveland Indians baseball team. Nor does anyone seem greatly exercised by baseball's Atlanta Braves (whose fans perform the "tomahawk chop" at moments of special excitement), or by the NFL's Kansas City Chiefs or by ice hockey's Chicago Blackhawks.

But lower down the sporting food chain, Indian names and mascots are disappearing. It's six years now since Chief Illiniwek, bedecked in his elaborate headdress, did his last pre-game dance as mascot of the "Fighting Illini" basketball team of the University of Illinois. Across the country, the number of school teams with Indian nicknames has shrunk from about 3,000 to an estimated 900. But Redskins would be – pardon the phrase – by far the biggest scalp yet.

Not only do they represent the nation's capital. In a city famous for rancour and partisanship, the Redskins are the great uniter, cutting across divisions of race, class or party. Rich or poor, black or white, Republican or Democrat; open their hearts when they die and you will probably find an Indian's head, with two feathers at its back, engraved in the team's colours of burgundy and gold.

Business-wise, too, the franchise has been a gift that keeps on giving. The Redskins haven't won a Super Bowl since 1992 but, according to Forbes magazine, they rank as the fifth most valuable sports franchise on the planet, behind only Manchester United and Real Madrid, baseball's New York Yankees and Washington's NFL arch-rival, the Dallas Cowboys.

But finding a new name may not be easy, as neighbouring Virginia found in its search for a new official anthem. In 1997, the state dropped "Carry Me Back to Old Virginny", harking back to Confederate times, because of the lyrics' reference to "massa" and "darkie". Sixteen years later, as far as I know, they still don't have a replacement.

And there's another problem with the Redskins. Any new name will have to fit into the team's own anthem, sung after every touchdown. "Hail to the Redskins," it goes, militaristic and maddeningly catchy, "Hail Vic-tor-y/Braves on the Warpath/Fight for old DC."

So where do you go? Some favour plain "Skins", as the team is familiarly known, and which also neatly alludes to the pigskin from which the football itself is made. Alas, in the first line of the anthem, "Skins" just wouldn't scan. Others suggest Washington Potomacs or Washington Piscataway, names taken from two Indian tribes once in the area.

Then there are the boring names, like Federals, Generals or Senators, referring to the city's main industries of government and war. And talking of local industry, did someone mention the Washington Lobbyists? My favourite suggestion is the Washington Drones, highly topical but, I fear, too Wooster-esque for the macho NFL. Which leaves the eternal standby: the Washington Warriors. It's kind of Indian-related, and it scans.