Back from stuffing myself for the holiday and can sit down to write a response.
corbon, I'm not sure what part of the king/vassal relationship you are not getting, but I've tried multiple times to put this context into this discussion. Robert is Ned's king and has been since the coronation following the sack of King's Landing. At the point when the conversation on the barrows takes place, Robert is the only person who can compel Ned to answer. Please note how the chapter starts with, "The summons came in the hour before the dawn ..." (AGoT 91) Only the king or the King's Hand can summon the High Lord of the North. With Jon Arryn dead, and Ned named his replacement, that means Robert is the only one who can compel Ned. He does so when he summons him, and he has the power to do so if he wants to get an answer on any topic he wants Ned to speak about.
What we see "live on screen" is Ned pleading with Robert to go no further in his questions on the topic of Wylla. Ned uses his supposed "dishonor" of himself and Catelyn, and Robert's love of him, to ask the king to stop. Robert answers in part with, "Well, I'll not press you if you feel so strong about it ..." but the point here is that Robert can press him on this point if he wishes to do so. Not that Ned can shut him down anytime he wants to. As Ned's king Robert can press him, can compel him, to the point of death if he wants, and Ned's only recourse is to die or forsake his vows and rebel. Now, I'm not saying that is likely in this moment, and I think Ned knows it isn't likely, but that is the power relationship here. It is not a conversation of equals, it is not just a conversation of old friends just talking about old times. That is part of the context you seem to want to leave out.
If you really think Ned believes he can manipulate Robert to do whatever he wants, the let me remind you of two important points to place this into context. Ned knows of Robert's hatred of all things Targaryen. It is what came between them when Robert accepted Tywin's "tribute" as so much "dragonspawn," and it is certainly evident in its full throated fury later in the same conversation on the barrows when the subject of assassinating Daenerys comes up - "Ned did not feign surprise; Robert's hatred of the Targaryens was a madness in him." (AGoT 93) But if Ned believes he can "shut Robert down" anytime he wants to, he is quite personally disabused of the idea very shortly thereafter when Robert judges Arya instead of allowing Ned to do so, and when Robert condemns Lady to death because he doesn't want to deal with his vile wife and son.
As to reasons Ned cannot tell Catelyn and Jon, I gave you reasons, but let's boil it down to this. Ned cannot risk having Catelyn or Jon looking into Wylla's history and identity because the more that is known, by friend or foe, risks word getting back to King's Landing of answers that can put Ned's story into doubt. As you point out, neither Catelyn or Jon, can compel Ned to answer anything. It is best for everyone's safety he doesn't and leaves the name of Wylla between Robert and himself.
As I said before, Robert accepts Ned's story because it fits a narrative he likes, and because he has nothing to make him think Ned is lying. Robert loves Ned being more like him with his lust towards common women. But that is why Robert so readily accepts Ned's tale, not why Ned has to tell him about Wylla being Jon's mom. Ned has to tell Robert something to divert Robert's attention away from the fact he is bringing Jon home with him after finding his dying sister. He cannot risk being compelled to answer in detail about Jon, so he has to have a cover story to tell Robert, or Jon Arryn, if one of the two of them wants more answers, and suspects the truth.
The pain he inflicts on Catelyn and Jon by refusing to talk about Jon's mother is part of the price Ned pays for his lies he has to live. There is nothing irrational about doing so. The secret of Jon's mother's identity has to be kept for the safety of all, and Ned does so for very logical reasons and fears. That is, if as you and I agree, R+L-J is likely, and Wylla is not really Jon's mother.
I raised the fisherman's daughter's tale because I've detailed my reasons in the past for thinking it is connected to Wylla's story and not just a new completely unrelated possibility Martin decided to throw out in ADwD. We don't have to go over those reasons again if you don't want to, or you just disagree. Suffice it to say I disagree with your certainty that it's only a local story with no connection to Wylla.
Ned can try to shut down any questions, but given more information, still more information will be sought. Or do you really think that because Cat no longer hears rumors of Ashara, that means Ashara's name never was discussed in Winterfell again? If so you have greater faith in Ned's powers to change human nature than I do.
To the last, you assume there is no one in charge of Ned's security, or to Ned's personal needs, and no people who are regularly assign those duties. I think it likely the opposite is true. Who knows if they are all dead before the rebellion ends, but, if not, there are going to be people who doubt the story of Wylla. It's the doubt that eats at heart of the story, and why Ned needs to avoid these question altogether.
If he names her she is no longer a "completely anonymous nobody " - she is a woman with a name. A woman with a name is far easier to trace than a completely anonymous nobody.
My point was not what you make it out to be. I have, from the beginning argued Ned's answer has to be put into the context of the entire series of questions and statements, including the clarifying question that ends them "you know the one I mean. Your bastard's mother?" It has been your argument that we can ignore parts of these statements and questions Robert uses to make sure it is clear whose name he is asking for. In fact, you have come up with a theory that Ned's response is to something not at all referred to in Robert's remarks - a wet-nurse from a long ago conversation.
So, let's clear this up.
1- Robert starts this with the question "what was her name, that common girl of yours?" but he doesn't leave it there and Ned doesn't answer him yet.
2 - Robert follows this up with attempts to come up with the name himself - "Becca?" "Merryl?" and "Aleena?" all of which Robert himself rules out as the name he is looking for. Ned doesn't respond.
3 - Robert gives Ned a clarifying statement, "you told me once" by which we know there was another earlier conversation in which Ned's common girl was named. Still no response from Ned.
4 - Robert then makes it very clear that he thinks the common girl Ned had an affair with is the mother of Jon by ending his series of questions with the question "You know the one I mean, your bastard's mother?" For Robert, and for the reader, and most importantly for Ned, it is very clear Robert is asking for the name of Ned's common woman lover, who is Jon's mother, that Ned once told him about.
5 - Ned then responds with the statement "Her name was Wylla." Once again, Ned's answer has to be viewed in the context of all of Robert's statements and questions. As a whole they make it very, very clear who Robert is asking about.
To read it otherwise, makes Ned into something he is not. I think I used the word "pettifogger", but you can call it what you will, it is not in Ned's character to play word games to hide this secret from his king. He instead has come up with what I think is a lie, and I think you agree with me on that, to cover just who is Jon's mother. It is simple and straightforward. Ned answers Robert with a lie. He doesn't dodge his questions or trick himself into believing he is answering a question from fifteen years pervious. He doesn't try to ignore some of the questions in order to come up with an answer that refers to someone else than Robert is asking about. He simply lies by supplying Wylla's name to Robert's questions.
One more thing on context. This is the first place we read Wylla's name. Previous to this we are led to believe Jon's mother is Ashara Dayne. The function of this dialogue for the story is to cast doubt on those rumors Catelyn hears and let the reader know there is another explanation that is supported by Ned's own statement and what Robert seems to know. This is where we get the idea that there is a mystery concerning Jon's mother's identity. We are supposed to read this and wonder if Ned is telling Robert the truth, and if not to ask why he lies to his best friend and king. We are not supposed to manufacture conversations from the past we have no basis to think happened.
Again, Robert's reach is not what the Targaryen's had in Dorne. Until Jon Arryn travels to Sunspear they are in fact in a state of war with the usurper on the Iron Throne. After that Robert never goes to Dorne and his influence is vastly diminished by the willful inability to bring justice to Elia and her children's killers. So, the Daynes can provide some measure of protection to Wylla from seizure.
More importantly, to the question of protecting her from a spy, they have the same ability as Ned to restrict those who get to their family. No "heinous isolation programme" is needed. Just the same protection given to family members and those who are with them. I assume there are guards providing family protection, nothing more. The control of the household's security that we would expect, nothing more other than including Wylla in those security measures.
The real difference is that in Starfall, Wylla lives there and is known as Jon's mother. In Winterfell she can't be and reside with Cat. Once again, to bring Jon home and raise him as an equal among Catelyn's household is one thing. To bring a woman known to be his mother and ex-lover of Catelyn's husband is an altogether new level of insult to Cat. So, no, unless Wylla is not known as Jon's mother, she cannot be brought to Winterfell. Not without reading a much darker, crueler side into Ned's character than we have reason to believe.
By leaving Wylla in the control of the Daynes, Ned allows himself the story he needs to be able to tell Robert, provides security for Wylla, and makes it possible that he can have a quasi normal home for Jon to be raised in. It is also an explanation that works with the known facts in the story. Wylla does indeed live with the Daynes. We don't know that she ever lived in Winterfell. She is someone the Lord of Starfall is on close terms with - he knows she nursed him and he knows intimate details of her story such as who he thinks is her son. In Winterfell we don't know of anyone outside of Ned who even knows who she is. With Ned, the only person we know he speaks of Wylla about is Robert. Why these facts would support the idea Wylla was in Winterfell is beyond me.