I think it was the sword:

"He had taken to wearing the sword immediately. It felt right beneath his ?ngers. He had told no one, not even Min, that he had recognized the weapon. And not, oddly, from Lews Therin’s memories—but Rand’s own."

Notice he recognizes the "weapon," not the scabbard. Also, notice it says he "recognized" the weapon, meaning he has seen it before, not just read about it. I guess that leaves something involving Heroes of the Horn or the ter'angreal in Rhuidean, as others have speculated, although I can't recall any swords of note seen in Rhuidean, and the only sword that is explicitly mentioned during the scene with the Heroes of the Horn is Artur Hawkwing's sword, Justice. Maybe, if it is Hawkwing's sword, it will help him get a truce with the Seanchan?

"He relaxed his hand on his sword, though he did not release it. He ?ngered the cloth-tied hilt. The weapon was long, slightly curved, and the lacquered scabbard was painted with a long, sinuous dragon of red and gold. It looked as if it had been designed speci?cally for Rand—and yet it was centuries old, unearthed only recently. How odd, that they should ?nd this now, he thought, and make a gift of it to me, completely unaware of what they were holding...."

The weapon is "centuries old," which seems to indicate it was not made during the War of Power. The only reason I can think of why it "looked as if it had been designed specifically for Rand" is because of the Dragon on the scabbard, but maybe whoever gave it to him just dressed it up with some paint, and that comment on being "designed specifically for Rand" has to do with how well the sword fits him. I don't think this will remain a mystery for much of the book - it seems unreasonable for Rand to know something about the sword (or the scabbard) and never think about it during one of his POVs, especially if he has the sword on him at all times.