The bean sprout first comes up in conversation when Lina asks Clary if Ember is really the only light in the dark world. Clary pulls the bean out of her pocket and says that something inside it knows how to make a bean plant. This might be some kind of clue as to what is going on with Ember, because Ember is also alive, and the life inside it had to come from somewhere.
Clary gives Lina the bean seed, along with a pot filled with some dirt, saying, "Stick the bean in here and water it every day […] It looks like nothing, like a little white stone, but inside it there's life. That must be a sort of clue, don't you think? If we could just figure it out" (4.98). Lina takes it home and waters it, and when Clary stops by to help Lina with the instructions, she points out that it's started to sprout.
Is it a coincidence that the bean starts to sprout right when Lina makes headway in deciphering the instructions, and learns that they're instructions for how to exit Ember? We don't think so. Just as the bean sprout breaks out of it seed shell and begins to flourish, Lina begins to figure out how the people of Ember can leave their long-term home and find a new, hospitable place to live.