What is significant about Winston waking up with “Shakespeare” on his lips?

There are a few reasons to consider here. One is that Shakespeare represents literature of the old time when raw emotion, loyalty and love (in all its forms) were exalted. It also represents the sort of ideas the Party is trying to stamp out. There are also allusions to Hamlet when he remarks that the whole society is merely conditioned for thinking everything is good, "Why then 'tis none to you; for there is nothing either good or bad, but thinking makes it so..." (Hamlet)