In the first episode in Season 3 of Game of Thrones, Tyrion had a conversation with his father, Tywin, where he demanded ownership of Casterly Rock.
Given how he has generally been treated with scorn throughout the first two seasons, did he really expect Tywin to grant his request? What was his motivations in asking in the first place?
I think a better question might be "Why hadn't he asked it before?" To which of course Tyrion himself answers something like
I knew the answer before I asked. Eighteen years since Jaime joined the Kingsguard, and I never once raised the issue. I must have known.
Why does he ask it now? In the books, he is just barely recovered from the wounds he received in the battle of the Blackwater, and in the assassination attempt performed by Ser Mandon Moore of the Kingsguard. It is possible that his brush with death, his belief Cersei tried to kill him, his father's ingratitude, the unfairness of what people think of him, etc. makes him throw caution to the wind and once and for all ask the question he has been dreading to ask for the bigger part of his life. And of course, it turns out he was right in being afraid.
There is nothing in the books that indicates that he had a specific reason for asking. It is portrayed as a spur of the moment kind of thing:
Tyrion rose on unsteady legs, closed his eyes for an instant as a wave of dizziness washed over him, and took a shaky step towards the door. Later, he would reflect that he should have taken a second, and then a third. Instead he turned.
"What do I want, you ask? I’ll tell you what I want. I want what is mine by rights. I want Casterly Rock."
He's the only remaining direct heir for Tywin.
Jaime is ineligible by virtue of being in the Kingsguard.
Cersei, even if you discount the general gender bias against women, is married into a different house.
That leaves Tyrion through the normal lines of direct succession.
He asks this to understand just how deep Tywin's scorn for him runs. By knowing for certain that Tywin has no intention of ever allowing Tyrion to have what would normally become his birthright, Tyrion understands exactly where he stands in his father's eyes.
Don't forget that Tyrion just played a big part in the King's Landing defense. Even if Cersei took all the credit, and that they would have lost anyway without the Tyrells, he expects that his father will see his strategy with the wildfire was brillant. And Tyrion has been a good Hand of the King, despite Joffrey being the usual asshole. So, Tywin could have seen this and put his hate aside, giving Tyrion a title he highly deserves and giving Casterly Rock a good lord. That would have been a smart choice if you ask me but the old man was blinded by his unjustified hate.
I think there is something to Tyrion wanting to assess his position in his father's eyes. The blatant rejection to his request frees Tyrion up to act against the Lannisters, his family, in the future - potentially allowing Tyrion to manipulate his way into more power. After time as Hand of the King he wants the power/influence again.
My guess is that he wanted a reward for his role in the defense of King's Landing. It was the first time he did something noble not only for the city, but for his family. At that point, he probably felt worthy enough to claim Casterly Rock.
