A couple of pages back a few people mentioned their preferred, and/or most likely, alternatives to RLJ. I was a bit surprised that AD+L=J wasn't almost unanimous. Although I generally agree with the criticisms mentioned by @Kingmonkey and @SFDanny, I still think it presents the strongest alternative case on all fronts. At least cumulatively, if not individually.
Some disclaimers before I make a small case for AD+L=J: I believe RLJ. I do not believe AD+L=J. However, IF GRRM is pulling a grand prank on us, and RLJ is a red herring, then I do believe AD+L=J is the only alternative for a variety of reasons. Also, despite a few people writing posts and essays in favor of AD+L=J, I'm not sure the best, or rather the most complete, case has been put forward. At least not that I know of. I'll post some stuff below that I think has been missed. Again, at least as far as I know. However, please keep in mind that I am, or at least may be, aware that alternative explanations and/or interpretations exist for evidence I list.
Since I'm posting this, I'm obviously open to discussion, answering questions, clarifications, etc. However, I'm not especially invested in this idea, which means I'm not overly enthusiastic about defending it. And I've probably shortchanged some of my arguments by limiting most of my research to AGoT. Because again, I don't even believe in AD+L=J. I believe RLJ. Please keep all of that in mind as you're reading. The gist of my argument is that, regardless of how certain theories have been evaluated and judged, AD+L=J is the only real alternative.
Before diving into some of the reasons why I think AD+L=J is the strongest alternative, really the only possible alternative in my mind, I want to point out the basis for these recent R+L≠J cases. I've been a member of these forums for about 4.5 years now. As I recall it now, the bulk of R+L≠J arguments were based on a lack of evidence. In other words, the evidence for RLJ wasn't very convincing. On the contrary, the case for RLJ is very strong, to say the least. A point which is rarely contested these days. Thanks in part, I think, to the many RLJ regulars who answered the call over the years.
I said all that to say this: All of the XYJ essays are based upon the premise that RLJ is an intention red herring by GRRM, and ought to be evaluated through that lens. Now, I would argue that red herrings don't really work this way, since you can't mislead an audience that doesn't know it's being fooled. And for so long, so much of the audience was blissfully unaware of RLJ. Anyway... leaving all that aside, assuming that RLJ is a red herring--and despite what I just got done saying, it really has to be in order for R+L≠J to work--we might have good reason to answer some of the AD+L=J criticisms mentioned. About the lack of positive evidence, specifically something between Arthur and Lyanna.
Because RLJ would be a non-traditional red herring, and really constitute a fake out on a grand scale, we would have to assume GRRM would be going "all in" in his attempt to deceive the more skilled reader who susses out RLJ. Which would explain the lack of positive evidence for AD+L=J. And IMO, would mean that we shouldn't put a big emphasis on the aforementioned "positive evidence." So it would mean that GRRM couldn't exactly tell us that Lyanna met Arthur Dayne and was enamored with Dawn. But, he could lay the groundwork for such a scenario by repeatedly showing us that she was interested in swords. Because if you're a girl who is interested in swords, there's a fair chance you'd be impressed by Dawn. And I suppose one could even understand this interest in "swords" as a metaphor for an interest in someone whose house has a sword on its sigil. Maybe.
Right now I want to address some of the reasons why AD+L=J is really the best alternative. Some are basic, some are narrative, others are meta-narrative.
It fits in the timeline. That's big. Nothing has to fudged, or chucked out the window altogether.
Dawn. A special sword that seems destined to play a part in the series. In fact, it may be the legendary Lightbringer, or the original Stark family sword, Ice. Or both!
Arthur Dayne appears to be an homage to Arthurian legend--As is the ToJ, btw. Perhaps meant to emphasize the Arthurian theme here.--combining King Arthur and Sir Lancelot. Arthur Dayne is not the king, but his name is Arthur and he has a very special, unique sword. Plus, he's meant to be the finest knight of his day. Well, Lancelot was also famous for stealing a lady who didn't belong to him.
Jonny Sword. The sword symbolism/theme with Jon. Even better if we're talking about Lightbringer. After all, it's not a large stretch that Dawn was the original LB. If you're interested in this line of inquiry I'd recommend reading @Schmendrick's R+L=LB essays. Lots of Jon-sword stuff. For AD+L=J, the key takeaway from the symbolism/theme of Jon and swords is the Dayne sigil, which features a sword and falling star. Now, because there is also a star, it's not as clear cut as when we say that lion symbolism means Lannisters, dragons for Targaryens and so on. But, if we're meant to think of sword symbolism the same way we do with lions and dragons, then I'm assuming that swords means Dayne. Alternatively, KGs are called "white swords" repeatedly throughout the series.
A bit further out there, but based on the similar wording of the LB and Dawn origin stories*, I wouldn't rule out the possibility that the sword and star on the Dayne sigil are representative of a man and woman. I'm sure we can figure out which is which. In other words, "sword" could symbolically refer to a male Dayne. This last part is a bit crackpot, but many moons ago I wondered if star:Stark::day:Dayne, which might play into that idea.
*(LB was forged in Nissa Nissa's heart, Dawn was forged from the heart of a falling star.)
Now, with that in mind, I'd ask everyone to (re)consider a few passages through the lens of AD+L=J. Some of them are more familiar than others.
He gave her a half smile. “Bastards are not allowed to damage young princes,” he said. “Any bruises they take in the practice yard must come from trueborn swords.” - AGoT, Arya I
This is popular with the RLJ crowd. Most of us take this to mean that Jon is trueborn since we know Joff is a bastard. Why does GRRM choose "trueborn sword" here? Is GRRM telling us Jon is a bastard sword Dayne?
He is not my father. The thought leapt unbidden to Jon’s mind. Lord Eddard Stark is my father. I will not forget him, no matter how many swords they give me. Yet he could scarcely tell Lord Mormont that it was another man’s sword he dreamt of… - AGot, Jon VIII
This is when Mormont gives Jon the (bastard) sword, Longclaw. Upon receiving this magnificent sword, Jon kind of freaks out and thinks about his father. For Jon that's Ned, but the audience knows better by now. The dramatic irony marks this passage as worth paying attention to, I think. "Knowing" that Jon's father is really Rhaegar, can we make sense of this paragraph, does it fit? It's tricky because the Targaryens don't have any special swords anymore. Blackfyre and Dark Sister are long gone. But, Schmendrick came up with an interpretation I quite like a couple of years back. That the "swords" are a reference to the IT.
Now, let's try the same thing with AD+L=J. "Knowing" that Arthur is Jon's father, can we make sense of this paragraph, does it fit? I think the answer is obviously yes. Jon's internal struggle here is about his father and his swords, specifically his father's sword. That's quite compatible with Arthur being Jon's father.
I'll post a few other passages from AGoT that stood out to me as possible evidence for the AD+L=J argument. They're interpretations of course, but I don't think any of them "interpret" more than normal. Nor do I think they are very convincing. Especially individually.
[Ned] lifted the greatsword high above his head.
Bran’s bastard brother Jon Snow moved closer. - Bran I
A greatsword, like Dawn, in the last sentence of one paragraph, and then the Jon is described as a bastard in the first sentence of a new paragraph. Meh. This barely caught my eye. But it's a possible connection between "greatsword" "bastard" "Jon." It gets better.
Jon swelled with pride. “Robb is a stronger lance than I am, but I’m the better sword, and Hullen says I sit a horse as well as anyone in the castle.” - Jon I
"I'm the better sword." It could be something on its own. But there's also something of a sword/lance dichotomy. Which might be worth noting when remembering the HH tourney. Rhaegar was the better lance, but there's no way he way he was better with a sword that Arthur. And, not only is Jon the better sword, but he sits a horse as well as anyone in the castle. Sword + horse riding could be interpreted as evidence of AD+L=J.
The next three are all from Jon's third chapter.
The courtyard rang to the song of swords.
The opening paragraph of Jon III.
Not my mother, Jon thought stubbornly. He knew nothing of his mother; Eddard Stark would not talk of her. Yet he dreamed of her at times, so often that he could almost see her face. In his dreams, she was beautiful, and highborn, and her eyes were kind.
Jon thinking about his mother in the armory; the place swords are kept or made. So, Jon's mother + sword(s).
By the time Jon left the armory, it was almost midday. The sun had broken through the clouds. He turned his back on it and lifted his eyes to the Wall, blazing blue and crystalline in the sunlight. Even after all these weeks, the sight of it still gave him the shivers. Centuries of windblown dirt had pocked and scoured it, covering it like a film, and it often seemed a pale grey, the color of an overcast sky… but when the sun caught it fair on a bright day, it shone, alive with light, a colossal blue-white cliff that filled up half the sky.
Jon emerges from the armory and the sun comes out. BftD/Long Night imagery perhaps. Then the Wall is described as "alive with light," just like Dawn is in the ToJ chapter. It's also described as "blue-white," a combination of blue and white. Perhaps like winter roses and Dawn.
The next three are from Jon's following chapter, Jon IV.
Jon was showing Dareon how best to deliver a sidestroke when the new recruit entered the practice yard. “Your feet should be farther apart,” he urged. “You don’t want to lose your balance. That’s good. Now pivot as you deliver the stroke, get all your weight behind the blade.”
Jon III and IV both begin with sword practice.
Life at Castle Black followed certain patterns; the mornings were for swordplay, the afternoons for work. The black brothers set new recruits to many different tasks, to learn where their skills lay. Jon cherished the rare afternoons when he was sent out with Ghost ranging at his side to bring back game for the Lord Commander’s table, but for every day spent hunting, he gave a dozen to Donal Noye in the armory, spinning the whetstone while the one-armed smith sharpened axes grown dull from use, or pumping the bellows as Noye hammered out a new sword.
Swordplay in the mornings for Jon, and most of his afternoons were spent in the armory. Again, armories are where swords and other weapons are stored or made.
“The Old Bear’s no fool,” Dareon observed. “You’re certain to be a builder, and Jon’s certain to be a ranger. He’s the best sword and the best rider among us, and his uncle was the First before he…” His voice trailed off awkwardly as he realized what he had almost said. - Jon V
Jon is the best sword, and the best rider. Arthur was certainly a sword, whether as a Dayne or a KG, and Lyanna was a fantastic rider.
I'll stop there, but also note that Jon stays in the armory when he becomes LC of the NW in ASoS. Again, the idea being that armories are places where weapons--like swords--are stored or forged.