I have a headcanon that Darth Vader must have downloaded the security recordings from that cell block sometime between the rescue of the Princess and the destruction of the Death Star, but he didn’t actually get a chance to listen to them until after ANH.
Probably while he was spinning in space, even. It was just eating at him. What was Obi-Wan doing there? What was he up to? Who was the Force-sensitive pilot, and did he have anything to do with Obi-Wan?
And then he remembered he had the recordings on him. And not a whole lot else to do, since he was floating in empty space. So he played them.
And the first thing he ever heard his son say was, “I’m Luke Skywalker. I’m here to rescue you.”
He knew immediately what it must mean. That’s why he destroyed the recordings. That’s why he began hunting obsessively for Luke Skywalker’s Rebel base. That’s why he didn’t tell the Emperor.
That’s why he tried so hard to sound surprised and uncertain when the Emperor informed him about “the son of Skywalker.” As if he didn’t already know. As if he hadn’t been directing his entire life around that information for months. As if his greatest loyalty was still to his Master.
(And those words were playing in his memory again, as Luke staggered under his weight and they made their lumbering, gasping way through the trembling corridors of the Death Star. As Luke knelt over him and removed his mask, and there was only compassion and love in his eyes. As Luke insisted that he had to save his father. I’m Luke Skywalker. I’m here to rescue you.
"You already have, Luke.")