DISCLAIMER:Any theory that is posted is just in fun. I am posting these as a reader myself, imagining I'm in your shoes and what you would think. None of these theories have potential spoilers in them and can be taken how you want to take them. They may or may not be true, but that's up to you to decide.
Early in the book Samalah tells Gerold the story of Aramound the Archer; a wildling figure who apparently impregnated King James Forrester's daughter. Samalah uses it to argue that Gerold is just like her and her people, he has their blood.
Let's ask the question about who Aramound even is. He is a wildling supposedly, but what is the fun in that? There may be inaccuracies to Samalah's story, she refers King James Forrester - King tells us this story is set atleast 900 years ago probably more actually, considering i dont think James was King Forrester during the conquest of Aemon Blackfyre. The story of Aramound is a story, the wildlings probably recount their history through speech although there has been instances of them writing and drawing (via Allerax Blackfyre. She probably taught them how to write). So the story of Aramound has most likely gone through 9 centuries (if not more) of telling, from generation to generation, there is a high possibility of things being changed or altered or lost over the years. In that case I argue that Aramound is actually an literal Archer, from the family of Aeyron Archer. Yes Aramound the Archer may be a descendant of Aeyron Archer; he may be of House Archer of Sun's Champion.
My argument is that House Archer is a First Men house that existed for thousands of years, it says so in the appendix - 'One of the oldest Houses in Cephri who came over during the invasion of the First Men twelve or so thousand years ago'. House Archer most likely got their name from their talent of being amazing archers, similiar to naming mechanisms of other First Men houses like House Strong or Forrester. We know that the wildlings are descendents of the First Men so there is a link there that Aramound may be a descendant of an Archer, but I don't think he's a descendant, I think he IS an Archer; a person with the name Archer.
Aramound is named 'The Archer' of course but what if we take it literally to mean Aramound the Archer, Aramound Archer. In Jon Darklyn's chapters we find out that the mountain clansmen of The Northmen refer to people as 'the', i.e 'the Gerold'. Wildings also have clansmen such as the Teal Clan and the Haunted Clan, there also exists tribes. This leads me to suggest that the creator of the story was a clansman that reffered to Aramound as 'The Archer' when in fact his name was simply Aramound Archer. Over the years the meaning has seemingly been lost and people just believe him to be an archer.
We can assume that this story then might have to do with some history with King James Forrester and King Aramound Archer. It may be likely that the Dawnlands and The North had a war with each other in The North and Aramound was an Archer king. If we want to put more thought into it, the timeline for there to be a Forrester King and an Archer king present concurrently, can be from 12,000 BC to 1 AC, so anywhere from that timeline this could have happened. How could a wildling find out about this war and why would they make a song about it? Well since we have that 12,000 year time frame we know that the mountains that divided The North and the Lands-Beyond-The-Winter-Waste didn't exist. Therefore the origin of this story may have been before the mountains emerged, which means that a wildling ancestor would have seen this war and told his ancestors, who eventually grow up to be wildlings, so it does make sense for the song to be about The North and The Dawnlands.
Conventionally aswell Archer's typically have two syllable names - 'Aey-ron, La-ra, Trys-tane, Ad-han, Co-dy, Mar-con, Ei-za, Hec-tor, Hai-lee, Dae-ron', the only one that breaks the convention is Nymeria. Aramound's three-syllable name breaks the convention so perhaps he may not be an Archer, or maybe like Nymeria he breaks the convention.
I think the song can also have a meaning other than suggesting Gerold is a descendant of a wildling. If we take it like Aramound is an Archer then this means that an Archer King impregnated a Forrester daughter. When has this happened before? Aeyron Archer and Joanna Forrester. Aeyron The Archer impregnates a Forrester girl(Joanna) while she is hidden away from her father Lord Forrester, who comes looking for her. Now, granted, Ulthor actually impregnated Joanna BUT in universe people believe Joanna is the mother to Aeyron's son. So in world people believe Aeyron impregnated Joanna; when Lord Forrester finds Joanna again after she was supposedly kidnapped and she has a child with her - just like the story. But what does it mean about Aeyron? Aramound is an allegory for Aeyron, and Gerold is being told about him for a reason. Remember back in book one when Aeyron is told - 'Aeyron Archer the Sun’s Champion and the Winter’s Son must meet'. It seems to suggest that Gerold is the Winter's Son then. Gerold is being told about Aeyron to prepare for themselves to meet, both of them seem to have a big impact on the plot then.