The thought that immediately came to my mind was that the sword had formerly belonged to High Lord Turak. I wish I could go back and re-read that section right now to see if any descriptions were given, but I’m at work and so it’s not possible.

Why do I think that? Well, first let me say it’s just an early theory given the limited info we have to work with. However, there are a couple of points in favor.

1)It’s definitely just a sword, and not a 'angreal. Why? Rand’s own thoughts reveal this. He notes the following The loss of his hand, though it wasn’t his primary sword hand, would leave him vulnerable if he were to face a skilled opponent. Even with saidin to provide a far more potent weapon, his first instinct was for the sword. He’d have to change that. It might get him killed someday. I think this implies rather strongly that the sword is nothing more than a sword, and not a device of the one-power. Rand makes a very clear distinction between using the sword (at a disadvantage) and using saidin. If the sword were an 'angreal, it wouldn’t mean anything that he is still at a disadvantage while using a real sword because of his hand, because that would not be the method of use of an 'angreal! Plus, we already have Callandor. I don’t think we’re introduced to yet another sword-shaped 'angreal. Rand wouldn’t think when using Callandor "what a disadvantage this would be against another sword, being fragile glass and all and me missing a hand", because that’s not how it’s used!

2)Rand notes the sword was “centuries old, unearthed only recently”. The centuries old bit is a direct implication to the sword being a power-wrought blade. It’s the blade of a Blademaster. Unearthed only recently is a direct connection to the ashes of Falme, and the sword only recently being recovered from the destruction that was left behind.

3)How odd, that they should find this sword now, and make a gift of it to me, completely unaware of what they were holding. This (combined with point-2 above) strongly implies Rand was just recently given the sword, and was given as a gift by the locals, ie: Domani’s upon his arrival. “that they should find this now” in that Rand has not been around these parts since the time of Falme. Not a small coincidence. “Make a gift completely unaware of what they were holding...” It’s a Seanchan sword, used by the High Lord who oversaw the Seanchan invasion of this area of Randland. So, yeah, that’s kind of “odd” that they should use an enemy's weapon as a gift of honor to the Dragon Reborn, their “savior” from said enemy.

4)The majority of the chapter leading up to this point was Rand ruminating over his need for peace with the Seanchan. He notes the attempts he’s made to entrust Seanchan peace, and seems rather put-off that the Seanchan do not acknowledge such attempts. And now he’s given a power-wrought blade, the late owner being one High Lord Turak, by the very people who the Seanchan have worked at controlling, and this blade may help to achieve the Seanchan peace if returned with respect.

So that’s it for now. It may be that a simple re-read of the Falme incident would dispel this theory, I’ll find out later today. Either way, for now this is what I’m going with over the other options being discussed.