There is a swordsmith couple, Gan Jiang and Mo Ye, from the Spring and Autumn Period of ancient China that has a legendary pair of male-female swords that they had supposedly forged that were named after them. The stories go that the King of Wu ordered Gan Jiang to forge a pair of swords for him, however Gan Jiang found he could not properly forge the metal until his swordsmith wife assists him in some way, either by giving up parts of her body or throwing herself into the furnace to infuse it with human qi. Either way, Gan Jiang hides the male sword and ends up angering the King, who has him killed and in a Hamlet sort of scenario, everyone and their son ends up killing each other.

These swords are only mentioned in these ancient texts so there's no proof they actually existed, but it does highlight that there could be certain swords of special import in ancient Chinese culture. For a sword that could quite possibly fit your imagination of being special, I'd point to the Sword of Goujian, an astonishingly well preserved sword also from the Spring and Autumn period, over 2000 years ago. It doesn't have a "name" per se, but thanks to the condition of the sword, eight characters are clearly visible on along the flat of the blade, "越王勾踐自作用劍" which translates to "Made for the King of Yue, Gou Jian's Personal Sword".

I mean, usually anything metal from that period looks like this. But thanks to the Sword of Goujian's craftsmanship and it's air-tight scabbard, it's survived all the way u until now.

I only have one named sword that I know anything about, so I'll tell you what I know. Most of this is taken from The Lone Samurai by William Scott Wilson.

The Drying Pole was a nodachi katana most famously wielded by a samurai named Sasaki Kojiro (~1580's-1612). The sword itself was made in Bizen Province some 300 years before being owned by Kojiro. It was unusually long for his time, as standards for katana length had changed throughout the years. Many swords were actually shortened to conform to these changing standards. The sword was named for this unusual length, supposedly looking like the long, curved bamboo poles used by the Japanese to dry their clothes on.

Kojiro himslef was regarded as one of the best swordsman of Japan at the time, and was particularly feared/respected for his use of the the "Turning Swallow Cut". The finer details of the technique Kojiro created are lost, but we know that it mimicked the flight of a swallow, sweeping downward before quickly striking again upwards. Despite the length and weight of his blade, he was known to be very precise and quick, and the move itself was inherently hard to block, making him a very formidable opponent.

In 1612, a fight was arranged between Kojiro and a man named Miyamoto Musashi, a ronin who had begun to make quite a name for himself as a swordsman. Musashi (very much on purpose) arrived to their duel on Ganryu island several hours late. Being stood up by some country peasant like Musashi infuriated Kojiro, and even worse, Musashi supposedly carried with him a large wooden sword (bokken) he carved en route to the duel. Musashi would not even show Kojiro his blade, a serious sign of disrespect. Kojiro took the bait; in a rage he walked into the surf to meet Musashi's boat and drew his sword, discarding the scabbard in the water. Musashi gave a final taunt to unbalance his opponent, stating that his treatment of his scabbard was surely evidence that he had already lost. Kojiro attacked rashly, forgoing his favorite technique. He struck directly downwards at his opponents head. Simultaneously, Musashi attacked with his longer weapon, landing a blow on Kojiro's head. Stunned, Kojiro fell to the ground. Musashi then promptly dealt a killing blow by caving in his chest, bowed before the lords in attendance, and left in the boat he arrived in.

Presumably, the sword was recovered after the duel by Kojiro's lord who was in attendance, but I don't know the fate of the sword, unfortunately.

Does your question include swords used as regalia, such as Joyeuse for the Kingdom of France ?