The part about Rand remembering the sword from his own memories has been discussed a lot so I won't redo it here.
One part I've not seen mentioned yet (and please forgive me if it has) is one of the snippets of writing we get at the beginning of books, this one at the beginning of tEotW after the Prologue.
"...And men cried out to the Creator, saying, O Light of the Heavens, Light of the World, let the Promised One be born of the mountain, according to the prophecies, as he was in ages past and will be in ages to come. Let the Prince of the Morning sing to the land that green things will grow and the valleys give forth lambs. let the arm of the Lord of the Dawn shelter us from the Dark, and the great sword of justice defend us. Let the Dragon ride again on the winds of time."
Emphasis is mine, and it's a super long quote for that little bit, but I wanted to give the context.
I started my reread really late and it made me kind of happy because I was reading this right when the first chapter of tGS was released. So it just makes sense to me that the sword Rand has is Justice and I would be very surprised if it turns out not to be.