Who is Quaithe? There really is far too little about her to come to any definite conclusions. It has been suggested that she might be a time travelling Melisandre, Sheira Targaryen, Ashara Dayne, Melisandre's mother, or Dany from the future.
The theory that she is Ashara Dayne is one I find convincing. First, she is mentioned more often than Shiera in the story. Also, she is more in the age range for the characters. There is no chance o time travel of any kind in the story. There is the possibility that Quaithe is none of these, but then there is nothing we can use to understand her motivations.
If Ashara is Quaithe it begs the question: why she is helping Daenerys? She's certainly trying to help her. She keeps interfering in Dany's life to give her unwanted and frequently unwelcome advice. Just when she began this practice is not at all clear. Dany has had strange dreams since she and her brother left Illyrio's manse.
Some have speculated that these have been sent by Bloodraven. I have difficulty believing this because there is no indication that Bloodraven has ever taken any notice of Dany's existence. My own conclusion is that Quaithe has been actively working to aid Daenerys. Even though the story gives every appearance of the two meeting for the first time in Vaes Tolorro. Yet she already knows a great deal about her. Some would point out that Quaithe has a glass candle. If true that actually explains how she could also invade Dany's dreams.
Why? Why is she so determined to aid Daenerys? The answer I come up with is: she isn’t, at least not primarily. I think her main motivation is helping Jon. I believe the true reason Ned went to Ashara after the Tower of Joy was to ask her to adopt Jon. This is a better reason than just wanting to return a sword. Ned did not want to dishonor his new wife, Catelyn, with a bastard. He also knew she had lost her own child. He might have reasonably believed that she would welcome a replacement. She refused because a)she was mad as hell over Ned killing her brother and b)she believed this would interfere with the prophecy of the Prince that was Promised.
In order to understand why; as in the Dune novels, precognition is a double-edged sword. Knowing the future doesn’t necessary result in being able to alter it. Trying to alter it, or make it happen, can have disastrous consequences. Just look at what happens to both Cersei and Rhaegar. Each tries to interfere with prophecy in opposing ways. Both end up suffering for it. I believe that Ashara both knows of and agrees with the Prophecy of the Prince. But she understands, as Rhaegar did not, that any attempt to affect the prophecy, even in a positive way, would only windup hurting them. She knew she could not take in Jon or else the consequences would be dire. Chief among these was that Jon would almost certainly not have joined the Nights Watch, which was an absolute necessity.
She chose to adopt Jon even if only she knew of it. In many ways, she looks upon him as the son she and Ned never had, even though she has no connection to him by either blood or marriage. She ran away to Asshai to learn magic. Perhaps also to look for Shiera Seastar. She may even have found her. She has been watching Jon his whole life, although whether she has done more than that I leave you to decide on your own. She may not be able to interfere. That doesn’t mean she can’t mess around with those near him. After all, there’s nothing in the prophecy explicitly saying how it will happen or what happens to everybody else.
This is where the ‘dragon has three heads’ comes in. Ashara knows that Jon has to have two helpers. Being the loving mother she is, naturally she desires him to have the best help he can. Also like a doting mother, she wants her son to marry the best woman, or women, he can. Any woman who has had to deal with an over demanding mother-in-law will know what I’m talking about. Whether Ashara knew beforehand that Dany and Arya would marry Jon hardly matters. She wants them to and is willing to take steps to help them.
But remember, she understands better than anyone, except Bloodraven, the dangers of trying to mess with the future. A lesson Melisandre unfortunately never learned. She also understands that visions are misleading.
One of the great mysteries of the story is how Arya gets accepted and trained by the Faceless Men. I have always been aware of the fact that the Kindly Man knows more about Arya’s future than he lets on. We now know how. Quaithe almost certainly engaged the Faceless Men to train and protect Arya. Why they would do this I don’t know, but it should be pointed out that it is never revealed whether the Faceless Men only do assassinations or whether they also take other kinds of jobs. They would make excellent spies after all. And Quaithe once introduced herself as ‘no-one’. Could she be a Faceless Man? This has been suggested before.
What was between Ashara and Ned? The answer, surprisingly, is not much. Instead of Ashara’s baby being Ned’s, what if it were Brandon’s? Here’s my thoughts: Ashara and Ned fall in love at Harrenhal. Brandon knows this and, like a real dick, he decides to take advantage of it. He goes to Ashara in the middle of the night, pretending to be Ned. A lot like the way Lysa visited Littlefinger. Brandon was supposed to be a lot like Ned, if bigger.
Ashara only discovers the ruse the next morning after. She then discovers that she’s pregnant. This would explain why she kept it a secret. She likely hated Brandon for ruining any potential relationship she might have had with Ned and she couldn’t claim the baby as Ned’s because Ned wouldn’t go along with it. Plus Brandon was still alive. She definitely didn’t want to marry him, even setting aside all other complications. It would also explain why she is so helpful towards Ned, hoping after Brandon’s death that marriage with him was still possible.
But then her daughter is stillborn, she finds out from Ned himself that he is already married, and that Robert is now king. She tells Ned where to find his sister, only to have him return with news of her brother’s death and a baby.
Ned had made a promise to Lyanna to keep her baby safe. But Ned did not want to risk having the boy in Winterfell. So he went to Ashara to beg her to adopt the boy. An enraged Ashara refused. Only after leaving her did she change her mind. By that point it was too late.
How much she wanted to pretend that Jon was really hers, or may have hated him because of her brother’s death, we’ll never know. What we do know is that Jon was the closest she would ever be able to get to the life with Ned she had wanted. Ned might have loved Ashara to some degree. But he loved his sister more. People have wrongly assumed that all the feelings were on his part. It was actually she who was emotionally invested in their relationship. Regardless,the end result was that she adopted Jon as their son in her own mind and has been working to aid him ever since.
All of this is speculation of course. It is entirely possible that there is no connect whatsoever between Ashara and Quaithe. If so there is nothing we have to comprehend her motivations. I think there is more than enough reason to believe that Quaithe is actively working to aid Jon, Arya, and Daenerys. Only if she and Ashara Dayne are the same person can I see any reason for why.