I agree with doma it is pretty clearly Lews Therin's sword. He could have found it and been told what it was in the portal stone lives, seen it in the periphery of his visions at rhuidean(less likely), could have found out about it in the ter'angreal in the stone(even less likely), could have had herid fel mention it to him in his research, or most likely it is part of the memories being blurred and now he remembers it as if it were his own. Which is why he says it is so odd.
I go with the last because, he wouldn't mention it being strange that it wasn't a memory from a man 3000 years dead if it were something built AFTER LTT died. Also, it wouldn't be strange for it to be in Rand's memories unless it SHOULD have been in LTT's.