It's explained in the very episode why Tyrion was sent, because he is family, Tywin's son. And of course he can't send Jaime because he's been captured. Note both Tywin's and Tyrion's first comments in the quote below. Tywin has realised Tyrion is quite good at ruling and King's Landing needs a good ruler at the moment. Tyrion's comment about strategy is also quite telling. He himself isn't the best at it so he wouldn't make much use in battle but Tywin is.
Tywin Lannister: They have my son! Get out, all of you. (to Tyrion) Not you. You were right about Eddard Stark. If he were alive, we could have used him to broker a peace with Winterfell and Riverrun, which would have given us more time to deal with Robert's brothers. But now – madness. Madness and stupidity. I always thought you were a stunted fool. Perhaps I was wrong.
Tyrion Lannister: Half wrong. I'm new to strategy, but unless we want to be surrounded by three armies, it appears we can't stay here.
Tywin Lannister: No one will stay here. Ser Gregor will head out with 500 riders and set the Riverland on fire from God's Eye to the Red Fork. The rest of us will regroup at Harrenhal. And you will go to King's Landing.
Tyrion Lannister: And do what?
Tywin Lannister: Rule. You will serve as Hand of the King in my stead. You will bring that boy king to heel, and his mother too, if needs be. And if you get so much as a whiff of treason from any of the rest – Baelish, Varys, Pycelle…
Tyrion Lannister: Heads, spikes, walls. Why not my uncle? Why not anyone? Why me?
Tywin Lannister: You're my son.
Game of Thrones, Season 1 Episode 10, "Fire and Blood"
It's also worth noting that Tywin is Hand of the King at that moment in time. However, he is too busy at war to go back and lead at King's Landing. He was playing to the strengths of his family, Tyrion ruling and himself fighting and being a strategist.
Tyrion's thoughts on the matter are elaborated on a bit more in this scene. He sends Tyrion, and not an uncle of someone else, because Tywin has given up on Jaime. That means Tyrion is now his only son, in his head, and so he needs to rely on him more to carry on House Lannister.
"More than you know, Father," Tyrion answered quietly. He finished his wine and set the cup aside, thoughtful. A part of him was more pleased than he cared to admit. Another part was remembering the battle upriver, and wondering if he was being sent to hold the left again. "Why me?" he asked, cocking his head to one side. "Why not my uncle? Why not Ser Addam or Ser Flement or Lord Serrett? Why not a … bigger man?"
Lord Tywin rose abruptly. "You are my son."
That was when he knew. You have given him up for lost, he thought. You bloody bastard, you think Jaime's good as dead, so I'm all you have left. Tyrion wanted to slap him, to spit in his face, to draw his dagger and cut the heart out of him and see if it was made of old hard gold, the way the smallfolks said. Yet he sat there, silent and still.
A Game of Thrones, Tyrion IX
