TBH I've always thought it was either Jaime or Cersei.
Littlefinger conspires with Lyssa to have her husband killed. A separate incident happens with Bran seeing Jaime and Cersei together. I thought it was clear either one of them would have him killed. Given what we know of the two, I'd hedge my bets on Cersei. When Caitlyn goes to Littlefinger, he deduces that this is a good time for chaos and decides to cause a further rift in the realm by claiming the Lannisters did it. That's just Baelish taking opportunity. Not only is Tyrion the brains of the family but 2/3 of his family absolutely hated him and wouldn't have remorse if he died, which blinds them from the fact he's the brains. Baelish, who knew Caitlyn, deduced that Cat would take Tyrion prisoner and the two families would fight. With the seeds planted, he conspires with Cersei to help kill Ned.
At least that's what I've thought. Littlefinger is a roach. Some things are totally his schemes, but for the most part he seems to be about climbing above the rubble and using it as his advantage more than anything. Although he's a schemer as well, obviously.