In my opinion, the content of the song the woman sings (this is in Part II, Chapter 4) is symbolic of what it is that Winston feels is missing in his society. That is, I think, why he thinks that the Proles might be the hope for the future.
I know the song was written by a "versificator" machine, but its lyrics express things that Winston wants. He wants true feelings like the ones the woman sings about. He wants memories that will last across the years and not get changed by the Party.
When the woman sings those lyrics, I think Winston understands that his hopes are in the lyrics of the song. (It never says this in the book -- this is my interpretation of why the song matters.)
In Part Two, Chapter Four, of 1984, Winston listens to a prole woman singing a song in the courtyard below his room in Mr Charrington's shop. While this song appears to have no relevance on the novel, it is, in fact, significant for two reasons.
First of all, the dialect of the song has some importance. It is composed in English and makes a refreshing change to Newspeak, the official language of Oceania. Newspeak is designed to make thoughtcrime an impossibility because it restricts people's ability to think negative thoughts about the party. In contrast, English remains expressive and emotive, even if the song has been composed by a "versificator" and does not technically make any sense. The song, then, harks back to a time before the party assumed power and this is, perhaps, the reason for Winston's enjoyment of it.
Secondly, the phrase, "opeless fancy," foreshadows the events to come. At this point in the novel, Winston is still uncertain about the safety of Mr. Charrington's shop. As he says in the next scene,
It was inconceivable that they could frequent this place for more than a few weeks without being caught.
In this understanding, it truly is a 'hopeless fancy' that Winston and Julia can ever be safe from the Thought Police. It is only a matter of time before they are arrested and imprisoned at the Ministry of Love where all hopes and dreams of rebellion will be eradicated, once and for all.