Paul is weak, but seems to have some insight into how things work and some appreciation for Hank. Perhaps his use of the expression, along with his knowledge that the Powers That Be may be out to get Hank, is to show that, weak and inconsequential as Paul appears, he does have some life outside the family and has picked up information, as well as the phrase. A residual decency lurks below the surface in Paul. He is defeated, not only by his society, but also by the women in his life, Hank's wife and mother. He has some sense of solidarity with Hank against it all. Hank himself is almost oblivious to his milieu.
From the incident with the trainman whistling the concerto, it is obvious that the people who know about Galt’s Gulch are still out in the world. They must be the source of “Who is John Galt?” However, it has always puzzled me that this particular meme is ubiquitous in the society and is a synonym for “Who knows” or “What's the use.” Perhaps it is just a tortured device Rand is using to establish the unifying theme and pique interest in Galt. As far as we know in the universe of the novel, Galt is unknown and his resistance is still underground. How would his name come to be a common expression? The only answer I can come to is that the members of the Gulch who are still in the world have been spreading the expression. There is fertile ground for cynicism and irony in a failing society where some can still sense that the problems are all manufactured. There is an obvious resonance of the phrase among even the most depressed.
I lived through the period when Rand was writing this. I was 14 when the book was published. I do not recall a lot of ubiquitous and current political catch phrases back then. Rand may be drawing on her experience in the USSR, when information/commentary was passed from person to person, not only because of the lack of mass communication, but the personal insecurity of publishing critiques of society/government. The usage may be to show that there is still some connection between individuals, even in a dehumanizing environment. Everyone, so far, recognizes the subtext of the phrase.
We find that out later in a flashback so long it almost takes up an entire chapter. It's when Dagny meets the man who accidently coined the phrase.