GRRM is great - I

When Arya is training with Syrio, one of the tasks he sets her to is chasing and catching cats.

"One by one Arya had chased them down and snatched them up and brought them proudly to Syrio Forel … all but this one, this one-eared black devil of a tomcat. "That's the real king of this castle right there," one of the gold cloaks had told her. "Older than sin and twice as mean. One time, the king was feasting the queen's father, and that black bastard hopped up on the table and snatched a roast quail right out of Lord Tywin's fingers. Robert laughed so hard he like to burst. You stay away from that one, child." - Arya III

While that does paint a funny picture for the reader, it doesn't make the cat relevant in and of itself, or does it?

Later, when Ned is in the black cells, he gets a visit from Varys. They discuss multiple things, all pertaining to Ned's situation. Then Varys drops this tidbit:

"Rhaenys was a child too. Prince Rhaegar's daughter. A precious little thing, younger than your girls. She had a small black kitten she called Balerion, did you know? I always wondered what happened to him. Rhaenys liked to pretend he was the true Balerion, the Black Dread of old, but I imagine the Lannisters taught her the difference between a kitten and a dragon quick enough, the day they broke down her door." - Eddard XV

And then it hit me; the black cat that Arya is chasing is Balerion. The sack of King's Landing happened 15 years ago, when Balerion was a kitten, meaning he'd be "older than sin" if he were still alive. Also, I think the fact that the story GRRM decides to present to us about the black cat is about a time he disrespected Tywin further compounds this, since it was in fact Tywin who ordered the death of Rhaenys. Not only that, but Lannister soldiers have had a heavy presence in the Red Keep ever since, which would explain why he is "twice as mean".