[Mithalwen]

Given how you fell about Rohan generally, you may not have read the section in Unfinished Tales about Cirion and Eorl. When Eorl rode from the North a mist came out of Lorien that sheltered them from Dol Guldur. While useful it rather collectively freaked them out. Also in one of the draft timelin. The Sons of Elrond fought with the Northmen (this was shortly after Celebrían's abduction and no doubt part of their vengeance), maybe they interracted enough with their temporary comrades to pass on some information on Grannie...though the star rereference is a bit indiscreet. Or they might have had dealings with Thranduil's folk who have a similar culture and in earlier times more contact with their southern kin.

I suspect, as with real world nursery rhymes and the like things are only part remembered and context is lost ..maybe the first few lines were familiar and Gandalf supplied the rest?