Daenerys is the daughter of Rhaegar and Ashara Dayne

[LynnS]

3 hours ago, MaesterSam said:

Hmm, why the Quiet Isle? How would she have gotten there? Don't they tell Brienne they don't normally take women? 

They have separate hut for men and women that can only be shared by married couples.  They do allow women to stay among them.

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A Feast for Crows - Brienne VI

"I want to stay with ser," said Podrick. "I mean, my lady."

"What you and Lady Brienne may do elsewhere is between you and the Seven," said Brother Narbert, "but on the Quiet Isle, men and women do not sleep beneath the same roof unless they are wed."

"We have some modest cottages set aside for the women who visit us, be they noble ladies or common village girls," said the Elder Brother. "They are not oft used, but we keep them clean and dry. Lady Brienne, would you allow me to show you the way?"

 

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A Feast for Crows - Brienne VI

"Yes, brother." Brienne unpinned her hair and shook it out. "Do you have no women here?"

"Not at present," said Narbert. "Those women who do visit come to us sick or hurt, or heavy with child. The Seven have blessed our Elder Brother with healing hands. He has restored many a man to health that even the maesters could not cure, and many a woman too."

"I am not sick or hurt or heavy with child."

 

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A Feast for Crows - Brienne VI

"Why do they call it the Quiet Isle?" asked Podrick.

"Those who dwell here are penitents, who seek to atone for their sins through contemplation, prayer, and silence. Only the Elder Brother and his proctors are permitted to speak, and the proctors only for one day of every seven."

"The silent sisters never speak," said Podrick. "I heard they don't have any tongues."

 

The Quiet Isle is also a place of smoke and salt;  the Saltpans and smoking beehives.  I'm inclined to think that Jon fits one of the smoke and salt prophecies and this is a place where noble ladies and/or pregnant ladies have visited in the past.  Although the brothers haven't always had a skilled healer until Elder Brother came along.