Who actually sent the assassin to kill Brandon Stark in Game of Thrones S01E02? Jaime believes it was not Cercei, and swears he did not do it and it makes no sense for him to lie to Catelyn about that since he admitted to her that he pushed him out of the window.
Great question. I am simply providing an answer so this question is no longer listed as Unanswered.
As stated, this question was already answered here on SciFi (as System Down mentioned in his comment). This is however based largely on information from the books, not the TV show, so it might be a spoiler if you plan to read the books:
Tyrion concluded after his own investigations that it was his nephew Joffrey who did it. Joffrey overheard his father (King Robert Baratheon) saying that putting Bran out of his misery would be the merciful and brave thing to do. Wanting to impress Robert, Joffrey stole the dragon bone hilt dagger, hired an assassin and gave him the dagger with orders to kill Bran.
Later, during a celebration (Joffrey's wedding I think), Tyrion strongly hinted to Joffrey that he knew all about his role in the assassination attempt. Joffrey's demeanor changed, which confirmed it in Tyrion's mind. But of course he never confessed.
There's also a scene in A Feast For Crows after Tyrion's escape, where Jaime and Cersei discuss it and conclude it was probably Joffrey.
It appears as though the show has left it ambiguous on purpose and hasn't explicitly stated who it was so far.
The evidence for each of the characters is the following:
The reason Bran is in a coma is because Jaime pushed him, maybe they sent the assassin to finish the job in case he remembered.
Jaime Lannister: I was thinking of us. You're a bit late to start complaining about it now. What has the boy told them?
Cersei Baratheon: Nothing. He's said nothing. He remembers nothing.
Jaime Lannister: Then what are you raving about?
Cersei Baratheon: What if it comes back to him? If he tells his father what he saw...
Game of Thrones, S01 E03, "Lord Snow"
Doesn't matter if the dagger was lost in a bet to Robert or Tyrion either of them could have got their hands on it.
He never owned the dagger along with the quote from Sansa when she's passing the sentence on Petyr we also know that Tyrion would never bet against his brother.
But that was another one of your lies. It was yours.
Game of Thrones, S07 E07, "The Dragon and the Wolf"
He has compassion for "Cripples, Bastards and Broken Things", specifically Bran in this case, as evidenced when he created the saddle design for him.
Tyrion Lannister: Then I'm not a dwarf. My father will rejoice to hear it. I have a gift for you. Give that to your saddler. He'll provide the rest. You must shape the horse to the rider. Start with a yearling and teach it to respond to the reins and to the boy's voice.
Bran Stark: Will I really be able to ride?
Tyrion Lannister: You will. On horseback you will be as tall as any of them.
Robb Stark: Is this some kind of trick? Why do you want to help him?
Tyrion Lannister: I have a tender spot in my heart for cripples, bastards and broken things.
Game of Thrones, S01 E04, "Cripplpes, Bastards and Broken Things"
He was "acquitted" during is Trial by Combat that was won for him by Bronn.
Although quite flimsy he appears to already have been aware of Valyrian steel when he is gifted Widows Wail. It can be argued that he knew about it from Ice but I believe this quote is to replace the similar event that happened in the books.
Careful, Your Grace. Nothing cuts like Valyrian steel.
So they say.
Game of Thrones, S04 E02, "The Lion and the Rose"
It appears as though Joffery did it in the books so it would make sense to be the same character, though we know the show likes to change things.
In the books it appears as though he sends the assassin to impress Robert as he says it will be a mercy to kill him than to live a cripple. It appears as though this doesn't happen in the show but the catspaw says something similar so could hint at this too.
Assassin: You're not supposed to be here. No one is supposed to be here. It's a mercy. He's dead already.
Game of Thrones, S01 E02, "The Kingsroad"
If Littlefinger did lose the dagger to Robert, as in the books, this could mean that the same person sent the catspaw, though I can't seem to find a quote to back this up.
Littlefinger, Petyr Baelish:
It was his knife at some point so it could have been his directly before the assassination attempt:
Petyr Baelish: Well well, this is an historic day. Something you don't know that I do. There's only one dagger like this in all of the Seven Kingdoms. It's mine.
Catelyn Stark: Yours?
Petyr Baelish: At least it was, until the tournament on Prince Joffrey's last nameday. I bet on Ser Jaime in the jousting, as any sane man would. When the Knight of the Flowers unseated him, I lost this dagger.
"Game of Thrones", S01 E03, "Lord Snow"
Chaos is a ladder: He likes to create chaos and then climb the ladder from it.
The wikia seems to state that as Sansa claims the dagger was his beforehand, see above quote, that he sent the assassin. The only way that they could know this is if Bran told them but as this hasn't been shown on screen we can only speculate if he did or not.
- This theory is quite tinfoily but it seems to suggest that because Bran sent the assassin himself when viewing the past to make sure he becomes the Three Eyed Raven.
Upproxx theorizes that the current Bran just wanted to ensure he would eventually become the Three Eyed Raven, which makes a lot of sense — we've seen the future Bran guide his younger self before, by taking the form of a raven in visions.
From a book canon perspective, however, we know that it was almost certainly Joffrey who sent the assassin as is worked out by Tyrion and later on Jaime/Cersei. He likely sent him to impress his father as some sort of mercy killing.
He remembered a cold morning when he'd climbed down the steep exterior steps from Winterfell's library to find Prince Joffrey jesting with the Hound about killing wolves. Send a dog to kill a wolf, he said. Even Joffrey was not so foolish as to command Sandor Clegane to slay a son of Eddard Stark, however; the Hound would have gone to Cersei. Instead the boy found his catspaw among the unsavory lot of freeriders, merchants, and camp followers who'd attached themselves to the king's party as they made their way north. Some poxy lackwit willing to risk his life for a prince's favor and a little coin. Tyrion wondered whose idea it had been to wait until Robert left Winterfell before opening Bran's throat. Joff's, most like. No doubt he thought it was the height of cunning.
The blade Joff chose was nice and plain. No goldwork, no jewels in the hilt, no silver inlay on the blade. King Robert never wore it, had likely forgotten he owned it. Yet the Valyrian steel was deadly sharp . . . sharp enough to slice through skin, flesh, and muscle in one quick stroke. I am no stranger to Valyrian steel. But he had been, hadn't he? Else he would never have been so foolish as to pick Littlefinger's knife.
The why of it still eluded him. Simple cruelty, perhaps? His nephew had that in abundance. It was all Tyrion could do not to retch up all the wine he'd drunk, piss in his breeches, or both. He squirmed uncomfortably. He ought to have held his tongue at breakfast. The boy knows I know now. My big mouth will be the death of me, I swear it.
A Storm of Swords, Tyrion VIII
Robert? Jaime had guarded the king long enough to know that Robert Baratheon said things in his cups that he would have denied angrily the next day. "Were you alone when Robert said this?"
"You don't think he said it to Ned Stark, I hope? Of course we were alone. Us and the children." Cersei removed her hairnet and draped it over a bedpost, then shook out her golden curls. "Perhaps Myrcella sent this man with the dagger, do you think so?"
It was meant as mockery, but she'd cut right to the heart of it, Jaime saw at once. "Not Myrcella. Joffrey."
Cersei frowned. "Joffrey had no love for Robb Stark, but the younger boy was nothing to him. He was only a child himself."
"A child hungry for a pat on the head from that sot you let him believe was his father." He had an uncomfortable thought. "Tyrion almost died because of this bloody dagger. If he knew the whole thing was Joffrey's work, that might be why..."
A Storm of Swords, Jaime IX
Regarding the show, rewatch the assassination attempt. It cuts from Cat's wounded hands to Daenerys having her wounded hands attended to. The back to back scenes in the show have provided many clues in the past. This suggests that it was Robert who did it as a mercy killing. He was probably drunk when he set it in motion.
I'm betting it was Bran trying to kill himself to change the past. Also Bran was not aware that his Mother was sitting vigil. He believes Catelyn had gone to Kings Landing, as she was not at WinterFell when he wakes from his coma. Which make this kind of a blind spot for the Bran, The Three Eyed Raven, since he believes he already knows what happened.
Actually Bran may not know that an assassination attempt was made on him while he was out after the fall. Would you burden a child who had already gone through what he had after he woke?
Also the Catspaw assassin seemed very surprised to see Catelyn in fact saying,"You're not s'pposed to be here; no one's s'pposed to be here. It's a mercy; he's dead already."
That last part is interesting to me "he's dead already." In a way the current Bran is dead, buried under the memories of all of Westeros. To him it is "Mercy Killing". He killing off the worst memories he has, his future.