Just relax, take it easy
You're still young, that's your fault
There's so much you have to know - Father and Son by Cat Stevens
Han's fatal mistake
Let's go back to that conversation Han and Ben in The Force Awakens were having on the Star Killer base just before Ben murdered his own father. It was two conversations happening at the same time, though only one party understood that.
I'm not going to call it a confrontation, but this fraught union was an all stakes, all in kind of game of Sabbac for Han. He perhaps just didn't realise what was at stake. I suspect he thought the stake was losing his only son forever, rather than the stake being his own life which meant he made a fatal mistake when he gambled.
When Han stepped out and yelled 'Ben', we knew it was on. By naming his son Ben, Han and Leia had accepted that their son Kylo Ren has a history that could not be ignored. He was a Skywalker after all, the grand son of Anakin Skywalker. It does not matter if his last name was Solo or Organna. All that matters is that they both loved their son and when Han went to him, he went purely as a father. Not as a man who had fought Ben's grand father many times and contributed in an indirect manner to his death.
When Kylo asks his father for help, I believe that Han misunderstood what was being asked of him. He assumes that Kylo is seeking his help to return to the light side of the Force. As a viewer, we are pushed in this direction. The sun is about to lose it's light as it's energy is drawn into the Star Killer weapon. The contrasts of blue and red which reflect the both sides are of the Force are shadowed around Kylo Ren. It's the big moment of the movie. We know the Kylo is clearly torn but it's not until he stabs his father though the chest that we realise he really is torn about being conflicted about letting the light of the Force into his heart and soul. Kylo Ren really wants to be the bad guy.
We know Kylo is torn as Kylo has spoken with Snoke who has told him he's about to face a pretty big test. But Han Solo does not know this. He thinks he's connecting with the Prodigal Son, the one who went off to party with the bad crowd but has now seeks his way home and needs some help from Dad to make sure mom is not going to smack him when he gets him.
The 'help' being anything that Kylo needs from his Dad. And so Han fatally offers him anything, doing anything thing for love but probably not expecting to die at the hands of his son. Even when he has had a red saber through his heart, Han lovingly strokes his son's cheek and looks at him. He gambled and lost, not knowing Ren was playing a different game of poker.
Ren's "Thank you" to his father as he falls is brutal. And unbelievably sad.