That's kind of spoilers. Although I suppose it doesn't overly matter...
It's heavily implied it was Joffrey, I suppose we'll never know for sure, but I'd say it's safe to consider that as canon. And apparently it's because after Bran got crippled the Baratheon family was all just chillin' out having a nice private family supper or some shit. Bobby B was hammered as per usual and was all like, "Fuck being a cripple would just be the single fucking worst thing, you'd be useless, they may as well put all cripples down, etc." Joffrey, who just admires the shit out of his "father" and wants to impress him, but who is also an idiot, and a sick fuck, then presumably takes action. Probably found the dagger in a treasury or some shit, or maybe it was a present once, who knows. Slaps together a terrible assassination plan. And that was that.
Meanwhile, Cat's like WHO WOULD DO THIS WHO OWNS THIS DAGGER, and Varys for once in his life is like "I dunno lol." So Littlefinger jumps on that shit. "Well in THAT case, I DO know whose it is! It's Tyrion's!" Sowing further discontent between the Lannisters and the Starks, further ensuring a breakdown at some point.
This. He's also the guy who cut off Jaime's hand, if you'll recall.