Why is it significant that O'Brien knows the last line of the stanza in the book 1984?

The first reason that O'Brien knowing this is signifcant is that it cements Winston's trust and liking of him.  He had been thinking about O'Brien for the entire book, wondering if he too felt the same way that he did about the Party.  When he gets the note from O'Brien, he is hopeful, and shows up at O'Brien's house.  Here, O'Brien reveals himself to be part of the resistance against the party, and him knowing the last stanza of that poem just cemented Winston's trust of him.  He liked him all the better for it--it proved to Winston that O'Brien knew better times, that he remembered the same things that Winston did, and it was a tacit admission that a different world had once existed.  It was a bonding point for the two of them, and gives Winston even more hope in the future.

Another way that him knowing this stanza is significant is that it is, if you really think about it, a significant piece of foreshadowing.  Winston had had conversations about that song and poem in the bedroom in the prole neighborhood.  He had recited the first parts of that poem out loud, and wished that he had known the rest of it.  In his rented bedroom, Winston had supposedly been safe from eavesdropping and the prying fingers of the Party.  O'Brien coincidentally knows the last stanza to this little, insignificant, obscure and totally random rhyme?  Is that coincidence?  Or did he somehow know of Winston's fascination with it and was prepared with it to win Winston over?  O'Brien's knowledge of it can be seen as too significant to be coincidence, and instead, evidence of foreshadowing of O'Brien's treachery.

One last possible way that it could be significant is that when we know who O'Brien is, we realize that he has access to whatever knowledge, information, research and power that he wants to.  He was able to look up and find an obscure rhyme.  His ability to find information that he wanted to is such a constrast from the rest of their society, where information is controlled down to the letter, and it shows just how power-hungry and hypocritical the Party is.  They can have access to information and truth, but their people can't.  They can drink fine wine and eat real chocolate, but their people can't.  They are all-powerful, and use that power to stay there.  O'Brien knowing the rhyme demonstrates the Party's power.

I hope that those thoughts helped a bit; good luck!