Why didn't the son get the inheritance? SPOILER

It was not explained.

I think it was because he was a barely coherent drug addict by that point. JPG was obsessed with the legacy of his company, Paul's mother had shown tenacity and great strength in adversity, and had a lot of his "steel", and qualities he saw in himself. The Dad was a loser, let's be honest. The moment he came into any kind of wealth he dived head first into a life of total hedonism. Plus, I imagine by leaving the inheritance to the mother this would mean Paul would eventually inherit it one day, and JPG for all his terrible flaws loved and respected that boy, in his own weird way.

Of course, things didn't quite work out for Paul, but they weren't to know that at the time of JPG's death.

Nothing went to the daughter in law, in reality. The son enherited 500 dollars, and the grandson was cut out of the will completely. Search "the real kidnspping of paul getty" and you'll find out a lot more.