We do not know who started the idea of poisoning Joffrey, Littlefinger or Olenna Redwyne (Queen of Thorns). That said the possibility of poisoning Joffrey was probably decided as early as ACOK when Littlefinger proposed the match to the Tyrells.
Olenna deciding to actually going through with it, was decided by Sansa confessing to Margarey and Olenna that Joffrey is a monster ...a bear, a bear all black and brown and covered with hair..., Ollena responding to Sansa calling Joffrey a monster, "that's a pity"