How did Genna Lannister disappoint Tywin?

Question

In A Feast for Crows, Genna (Tywin Lannister's only sister) and Jaime had the following conversation:

"Tired?" His aunt pursed her lips. "I suppose he has a right to be. It has been hard for Kevan, living all his life in Tywin's shadow. It was hard for all my brothers. That shadow Tywin cast was long and black, and each of them had to struggle to find a little sun. Tygett tried to be his own man, but he could never match your father, and that just made him angrier as the years went by. Gerion made japes. Better to mock the game than to play and lose. But Kevan saw how things stood early on, so he made himself a place by your father's side."

"And you?" Jaime asked her.

"It was not a game for girls. I was my father's precious princess . . . and Tywin's too, until I disappointed him. My brother never learned to like the taste of disappointment." She pushed herself to her feet. "I've said what I came to say, I shan't take any more of your time. Do what Tywin would have done."
_{AFFC - Jaime V}

How did she disappoint Tywin?

From what I can guess it could be Genna's promiscuity:

It was hard not to feel contemptuous of Emmon Frey. He had arrived at Casterly Rock in his fourteenth year to wed a lioness half his age. Tyrion used to say that Lord Tywin had given him a nervous belly for a wedding gift. Genna has played her part as well. Jaime remembered many a feast where Emmon sat poking at his food sullenly whilst his wife made ribald jests with whatever household knight had been seated to her left, their conversations punctuated by loud bursts of laughter. She gave Frey four sons, to be sure. At least she says they are his. No one in Casterly Rock had the courage to suggest otherwise, least of all Ser Emmon.
_{AFFC - Jaime V}

Tywin disapproved of such behavior and he would be very disappointed if he found a lioness of the Rock consorting with household Knights.

Or it could be that Tywin expected Genna to refuse to marry Emmon Frey but she ended up being persuaded by her Lord Father (Genna evidently never wanted to marry Emmon but for some undisclosed reason she did and even after they were wed, she never liked her husband) and therefore disappointing her lord brother. I can picture Tytos begging his daughter to save his honor by agreeing to the marriage and Tywin commanding her to protect the honor of the House by refusing. After all, on the wedding day, it was her choice that mattered, especially since her father was a weak man. If she chose her Father's honor, that would have disappointed her brother. Tywin was only ten when he objected to the betrothal, I doubt he would have given up without a fight when he was older and betrothal was turning into reality. But his sister choosing her father would put an end to any of his efforts.

She also angered Tywin once by saying that Tyrion was his true son, making him so angry that he didn't speak to her for half a year but I don't see how could that disappoint Tywin.

Has George ever commented on that other than "Keep Reading"? If he hasn't, "We do not know" or "No he didn't" are perfectly valid answers.

Answer 1

Although we can't be certain without proper confirmation, I've been led to believe that it's because of her marriage to Emmon Frey.

_{A lot of this answer is based on the blog post by Joanna Lannister. Full credit goes to her, including the parts that aren't written by her.}

We know that from a young age Tywin felt entitled to the Lordship of House Lannister and that his father, Tytos, was a useless oaf who failed to do his duties as Lord of a Great House.

Lets take the passage in it's entire context. (Emphasis mine)

“It was not a game for girls. I was my father’s precious princess … and Tywin’s too, until I disappointed him. My brother never learned to like the taste of disappointment." She pushed herself to her feet. “I’ve said what I came to say, I shan’t take any more of your time. Do what Tywin would have done.”

“Did you love him?” Jaime heard himself ask.

His aunt looked at him strangely. “I was seven when Walder Frey persuaded my lord father to give my hand to Emm. His second son, not even his heir. Father was himself a thirdborn son, and younger children crave the approval of their elders. Frey sensed that weakness in him, and Father agreed for no better reason than to please him. My betrothal was announced at a feast with half the west in attendance. Ellyn Tarbeck laughed and the Red Lion went angry from the hall.

The most interesting parts have been highlighted above.

Let us begin with "Precious princess". As Joanna Lannister says, whenever GRRM uses the term "Precious princess" he's talking about a younger girl, this would rule out Genna's 'misdemeanour' with household knights as the cause for disappointment.

Arya didn’t care about [Elmar’s] precious princess

Ser Arys will permit no harm to come to his precious princess

“An ear.” Cersei stared at him, aghast. She was just a child, my precious princess.

Genna, before the attached quote above, describes each of Tywin's siblings politically, when Jamie asks about herself she states this is "not a game for girls". Although Genna couldn't align herself politically with Tywin, her husband would've been able to. Tywin was aware of the opporunity that stood in front of him. Genna provided an excellent tool for creating an alliance (Similar to Roberts intention with his betrothal to Lyanna, and Brandon's to Catelyn), and an alliance of greater houses could've been formed, all of this, however, was squandered by Tytos and his weakness.

Some believe (from the above) the Roger Reyne left the hall in anger when Genna's betrothal was announced because he had intended the wed a Reyne into the Lannister household (as he had done with Ellyn Reyne). Tywin likely knew of this or something similar and realised he could've benefited greatly from this as he would've been able to turn Genna (and in turn her husband) into a great Asset. But instead of House Reyne, Tywin ended up with a lesser son of House Frey.

Although Genna isn't to be blamed for this betrothal, we know that Tywin is capable of laying the blame on people who had no choice in the matter, for example, Tyrion's dwarfism besides the fact Tyrion had no control over his gene selection and birth.

Tywin's distaste for Emmon Frey continues onwards, this is supported in the "rumour" paragraph

It was hard not to feel contemptuous of Emmon Frey. He had arrived at Casterly Rock in his fourteenth year to wed a lioness half his age. Tyrion used to say that Lord Tywin had given him a nervous belly for a wedding gift. Genna has played her part as well. Jaime remembered many a feast where Emmon sat poking at his food sullenly whilst his wife made ribald jests with whatever household knight had been seated to her left, their conversations punctuated by loud bursts of laughter. She gave Frey four sons, to be sure. At least she says they are his. No one in Casterly Rock had the courage to suggest otherwise, least of all Ser Emmon.

Joanna Lannister claims she has no reason to believe the sons aren't Freys, and neither do I. However, Tywin seemed to take pleasure in his abuse of Emmon Frey, and Genna aspiring to be like Tywin did the same. Tywin, and in turn Genna, seem to do what they can to ruin Emmon's life, with Tywin allowing Genna to be flirtatious with his house-hold knights, as if he'd felt this was brandishing the Lannister name, he would've ensured it ended immediately.

Furthermore, Tywin was aware this would have no effect on the Lannister name. Rumours that the weak Emmon was unable to control his wife, or that she'd given him horns (made a cuckold of him) would only look poor on house Frey. After House Frey weaseling their way into a marriage with a great house, Tywin ensures that he gets his revenge.

To conclude, _{as I know this isn't worded very smoothly, big text make dum dum go confused} although Genna did not (necessarily¹) have a choice in her betrothal, Tywin is disappointed that it went through, and although it's not her fault (entirely¹), as it says above, we are aware Tywin is capable of blaming those who had no outright choice in the matter, i.e. Tyrion's birth)

Answer 2

I have had a search around and found various other posts asking the same question. There's this westeros.org post and this reddit post but most of the answers just come to the same theories as you describe in your post.

It appears that the most popular theory is because she said Tyrion is Tywin's son not Jaime:

"Jaime," she said, tugging on his ear, "sweetling, I have known you since you were a babe at Joanna's breast. You smile like Gerion and fight like Tyg, and there's some of Kevan in you, else you would not wear that cloak . . . but Tyrion is Tywin's son, not you. I said so once to your father's face, and he would not speak to me for half a year. Men are such thundering great fools. Even the sort who come along once in a thousand years."
^{A Feast for Crows, Jaime V}

The only concrete evidence I have found is on the AWOIAF Wiki on Genna's page which states:

Genna had been her father's "precious little princess", and later her brother Tywin's as well. However, according to her, she fell out of Tywin's favor when she "disappointed him", although it is not known what Genna did.

So it appears that the answer is "We do not sow know".