This dialogue occurs in Empire Strikes Back:
Emperor: The Force is strong with him. The son of Skywalker must not become a Jedi.
Darth Vader: If he could be turned, he will become a powerful ally.
Emperor: Yes. He would be a great asset. Can it be done?
Darth Vader: He will join us or die, my master.
Did Vader ever truly intend to kill the 'son of Skywalker'? If so when did he change his mind? And what caused the change of heart?
My reading of this scene has always been that each of Vader and Palpatine are plotting to kill the other and replace him with Luke:
Both of them are well aware of what's going on here: each of them wants to depose the other and replace him with Luke, and both of them know this. "If he could be turned, he will become a powerful ally" - yes, but the Rule of Two then dictates that one of Palpatine or Vader is going to have to go - so it's a game for them.
Spoiler alert, obviously.
Maybe he has been blind for his son, but I think he tries to save him, as he is his father. If you read a little more of your quotation, it should become a little clearer:
Darth Vader: What is thy bidding, my master?
Emperor: There is a great disturbance in the Force.
Darth Vader: I have felt it.
Emperor: We have a new enemy, the young Rebel who destroyed the Death Star. I have no doubt this boy is the offspring of Anakin Skywalker.
Darth Vader: How is that possible?
Emperor: Search your feelings, Lord Vader. You will know it to be true. He could destroy us.
Darth Vader: He's just a boy. Obi-Wan can no longer help him.
Emperor: The Force is strong with him. The son of Skywalker must not become a Jedi.
Darth Vader: If he could be turned, he will become a powerful ally.
Emperor: Yes. He would be a great asset. Can it be done?
Darth Vader: He will join us or die, my master.
(as seen here)
His first answer is no "... and we should stop it", it's more like a "yeeees, it's been tickling, but, ack, forget it..." He, IMHO, underestimates him for purpose. He wants that his master won't believe that this little boy will ever be a menace.
Then again Vader pretends being dumb and has to search his feelings. Like threatening, force feelings and... "that's my son, he will be stronk".
After that Vader truly wants to save him while saying that a kid won't harm anyone. He is saying that Luke is innocent, no target.
The Emperor then states, that this kiddo is more than Vader wants to admit. That's were Vader only sees one option to save his rugrat: Turning him to the dark side. Because then Luke can live - and side by side by his father.
The Emperor thinks that'll be a good deal - the two strongest force users under his command. So he agrees.
The last sentance, IMHO, is more like a platitude. "I'm his father, he'll join me."
And in the end we see, that Vader himself turned to the other side. :)
As Vader obviously tries to save and/or convinvce him to the other side, Vader did not intend to kill his son. Killing would be a much easier option - if Vader had no feelings left.
Per Vader's inner monologues in the novel "Shadows of the Empire", he had no intention whatsoever of killing Luke. In fact, the major plot point of the novel revolved around:
Prince Xizor - head of Black Sun and a potential rival to Vader - attempting to kill Luke Skywalker in order to please Emperor Palpatine and stick it to Vader. He knew that Darth Vader specifically wanted Luke alive, whereas Palpatine wanted him dead (or turned, but Xizor didn't know that part). Vader had plans to turn Luke and make him his apprentice, but Xizor stood in his way. So, obviously, Vader decided to take Xizor out - both for his own benefit as well as Luke's safety.
Also, the novelizations of Episodes V & VI seem to indicate that Vader was indeed sincere when confronting Luke on Cloud City and again on Endor. During the conversation with Palpatine, Vader was simply telling him what he wanted to hear... a fact that Palpatine probably knew, but his SOP was to play things out and see what happens. From his standpoint, if Luke could be turned he would provide a younger & more powerful apprentice to replace the ailing and limited Vader (whose potential Force abilities were limited by his cyborg parts).
There was also the duel they had in Empire Strikes Back, there's a few facts regarding the duel that make it clear that Vader didn't want to kill Luke:
For starters, he specifically says at one point in the duel:
Vader: Don't make me destroy you!
He didn't want to kill Luke, only if he absolutely to.
Another point is that, at this time Vader is still stronger in the force than Luke is and could have easily just force-choked him anytime and ended the fight right there. There are several instances in the movies where Vader has no problem with force choking others who get in his way/anger him (even force choking his own wife in Episode III). He could have just force choked "Young Skywalker" when he was vulnerable and hanging on the platform in Cloud City during their conversation.
And lastly, he didn't seem to be fighting at 100%, he was mostly just blocking Luke's strikes and throwing things in his way to distract him. He may not have been at his full potential also due to age and the limitations of his suit, but as seen in the below video, he's also purposely missing Luke with his strikes a few times. Just hitting the wall or railing and creating sparks instead. Only at the end of the duel when Luke hits him does he feel the need to actually inflict harm on Luke if only to disarm him (no pun intended).
There were things in Vader that survived all his struggles with the dark side. Granted, there wasn't much but what did survive was what kept part of him shielded from even the powerful, all seeing and sensing Emperor. He loved his son and he had every intention of finding him and finishing his training. But this had to be hidden completely from the Emperor as did Vader's growth in power. Limited by the suit, a fail safe that would kill Vader as soon as the Emperor detected betrayal, Vader bided his time. He's moved by Luke's ability to build a lightsaber and his confidence to turn himself in like he did. Anakin realizes he has lost completely any chance at playing a part in Luke's becoming a man and a master of the force. But Vader schemes yet again, Luke may be powerful enough to kill the Emperor or at least distract him and Vader can kill the Emperor.
Vader's love of Luke is what kept enough of Anakin alive and still a "good" soul, amazingly the Emperor could not sense this good side to Anakin but Anakin's son could. This really impressed Anakin and Vader and he realized that letting things play out as they may might not be all that bad of an idea. The more he learned about Luke, the more AnaKIN realized the chances of the Emperor's survival were fleeting...a revenge dish he's been longing to serve for a very long time.
