Some House's proudly bear their ancestral weapon, usually a Valyrian steel made sword even if there are exceptions such as Dawn.
Which one of these ancestral weapons is claimed to be the oldest?
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Since your question is about all ancestral swords, not just the Valyrian steel ones, Dawn is oldest ancestral sword in Westeros. It is claimed to have been passed down to the greatest Swordsmen of the House for ten thousand years.
Ser Gerold says this about his house:
“There was an Arthur Dayne,” Myrcella said. “He was a knight of the Kingsguard in the days of Mad King Aerys.”
“He was the Sword of the Morning. He is dead.”
“Are you the Sword of the Morning now?”
“No. Men call me Darkstar, and I am of the night.”
Arianne drew the child away. “You must be hungry. We have dates and cheese and olives, and lemonsweet to drink. You ought not eat or drink too much, though. After a little rest, we must ride. Out here on the sands it is always best to travel by night, before the sun ascends the sky. It is kinder to the horses.”
“And the riders,” Spotted Sylva said. “Come, Your Grace, warm yourself. I should be honored if you’d let me serve you.”
As she led the princess to the fire, Arianne found Ser Gerold behind her. “My House goes back ten thousand years, unto the dawn of days,” he complained. “Why is it that my cousin is the only Dayne that anyone remembers?”
AFFC - The Queen Maker
The founder of House Dayne, according to the legend, tracked a star to the mouth of Torentine river and constructed a castle, Starfall, where the fallen star was discovered.
At the mouth of the Torrentine, House Dayne raised its castle on an island where that roaring, tumultuous river broadens to meet the sea. Legend says the first Dayne was led to the site when he followed the track of a falling star and there found a stone of magical powers. His descendants ruled over the western mountains for centuries thereafter as Kings of the Torrentine and Lords of Starfall.
TWOIAF: Dorne- Kingdoms of the First Men
The sword was forged most likely from the heart of this star. So by that assumption, if the founder of House Dayne forged the sword and the House has been around for 10,000 years, the sword is just as old.
Something his father had told him once when he was little came back to him suddenly. He had asked Lord Eddard if the Kingsguard were truly the finest knights in the Seven Kingdoms. “No longer,” he answered, “but once they were a marvel, a shining lesson to the world.”
“Was there one who was best of all?”
“The finest knight I ever saw was Ser Arthur Dayne, who fought with a blade called Dawn, forged from the heart of a fallen star. They called him the Sword of the Morning, and he would have killed me but for Howland Reed.” Father had gotten sad then, and he would say no more. Bran wished he had asked him what he meant.
ACOK- Bran III
In any case, TWOIAF confirms that it is likely that the sword has been in the family for several thousand years.
The Daynes of Starfall are one of the most ancient houses in the Seven Kingdoms, though their fame largely rests on their ancestral sword, called Dawn, and the men who wielded it. Its origins are lost to legend, but it seems likely that the Daynes have carried it for thousands of years. Those who have had the honor of examining it say it looks like no Valyrian steel they know, being pale as milkglass but in all other respects it seems to share the properties of Valyrian blades, being incredibly strong and sharp.
TWOIAF: Dorne- The Andals arrive
I don't think it's clear exactly how old most of the weapons are though we know Ice to be 400 years old.
"I am always proud of Bran," Catelyn replied, watching the sword as he stroked it. She could see the rippling deep within the steel, where the metal had been folded back on itself a hundred times in the forging. Catelyn had no love for swords, but she could not deny that Ice had its own beauty. It had been forged in Valyria, before the Doom had come to the old Freehold, when the ironsmiths had worked their metal with spells as well as hammers. Four hundred years old it was, and as sharp as the day it was forged. The name it bore was older still, a legacy from the age of heroes, when the Starks were Kings in the North.
A Game of Thrones, Catelyn I
However, most Valyrian steel swords in Westeros came there after the Valyrians took Dragonstone some 200 years before the Doom. As such it is most likely that any Valyrian steel sword in Westeros is similar in age to Ice.
With the destruction of the Rhoynar, Valyria soon achieved complete domination of the western half of Essos, from the narrow sea to Slaver's Bay, and from the Summer Sea to the Shivering Sea. Slaves poured into the Freehold and were quickly dispatched beneath the Fourteen Flames to mine the precious gold and silver the freeholders loved so well. Perhaps in preparation for their crossing of the narrow sea, the Valyrians also established their westernmost outpost on the isle that would come to be known as Dragonstone some two hundred years before the Doom. No king opposed them—and though the local lords of the narrow sea made some effort to resist it, the strength of Valyria was too great. With their arcane arts, the Valyrians raised the Citadel at Dragonstone.
Two centuries passed—centuries in which the coveted Valyrian steel began to trickle into the Seven Kingdoms more swiftly than before—though not swiftly enough for all the lords and kings who desired it. And although the sight of a dragonlord flying high above Blackwater Bay was not unknown, it occurred more frequently as time passed. Valyria felt its outpost was secured, and the dragonlords thus continued their schemes and intrigues on their native continent.
The World of Ice and Fire, Ancient History: The Doom of Valyria
The Wiki of Ice and Fire page on Dawn appears says the following of the sword and as such it would be the oldest.
According to legend, the founder of House Dayne tracked a falling star to the mouth of the Torentine, and Starfall was then constructed where a magical stone was discovered. The greatsword Dawn was forged from the heart of this fallen star and has possibly been wielded by Daynes for ten thousand years.
However, this appears to be sourced from the following quote which doesn't make it clear if the sword was around then or not. Still the sword is likely from the houses foundation and so it will be older than any Valyrian steel sword.
As she led the princess to the fire, Arianne found Ser Gerold behind her. "My House goes back ten thousand years, unto the dawn of days," he complained. "Why is it that my cousin is the only Dayne that anyone remembers?"
A Feast for Crows, The Queenmaker