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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..."
I think he was saying "Shakespeare" cause it was reminiscent to him of better times, certainly 'cultural' better times, Winston in essence was really a man who tried to escape the banalities of his life, which the party so much was oppressing towards its people, the party really did its best to extinct every kind of 'culture' out of the people's mind, Winston here really showed that he had a great power of Will, he wasn't and wouldn't never give away his culture, he also to my opinion was an optimistic mind, the fact that he woke up with the word "Shakespeare" must have gone together with a(n very) optimistic smile, I think, and still phantazise about that!