However, it also makes the most sense:

Justice is a legendary sword, which is centuries old. Rand saw Justice at Falme, while LTT doesn't know the sword from his memories. Rand might find it odd, that Justice is found now and given to him as a gift, because Rand is just trying to make an alliance with the descendants of Hawkwing, the Seanchan. When Rand goes to kill Graendal, he muses that he might wear the sword as a symbol of something; maybe of him delivering justice to a Forsaken? Besides that, there are Prophecies which link Justice (or at least a blade) to Rand. And it also seems that Justice is pictured alongside the Horn of Valere on the Westland map at the beginning of the WoT books, which could strengthen its importance.