I'm pretty sure he'll be mourned by more than just his family. For any person to control a country, you'd expected them to have the support of at least enough people to keep the government running, and it has, for decades. Plus, with the isolation and state control of the media, it's probably pretty easy to make a large portion of the population love their leader. That would seem to be true whether or not the leader is considered "good" (particularly by foreigners). I'm not sure how long lasting it will be considering the hardships faced by many North Koreans, but I would expect it to last quite a while, especially considering the family is still in control. Those who were too young to be taught to love their leader will also be too young to have learned to hate him. I suspect local sentiment is more likely to fade than reverse.
If you've gone looking for any news on the subject, you'll probably find pictures of quite a few people mourning. I do believe a fair amount of that sentiment is genuine. Particularly considering the political climate. I can also understand people thinking that mourning is misguided, but I don't believe that makes it any less genuine. Perhaps more in question is what kind of numbers are we talking about here? Many of those pictures are of course provided by the state run media, so I suppose that means take them with a grain of salt.
I believe I've heard of that documentary you were talking about. I've never seen it though. I heard after someone had surgery, the first thing they did was thank a picture of the leader on the wall. (I can't remember if it was Kim Jong Il, or his father though). Interesting reaction. The friend that told me about the documentary also went on about how the person was basically blind in the first place because the policies of that leader had prevented adequate medical care from being available that could otherwise have prevented his blindness. But nevertheless, it's an interesting reaction, shall we say.
I've seen a documentary about families between North and South Korea reuniting. It was interesting to note how one of the North Korean's kept thanking the leader for various things. (I also can't remember if it was Kim Jung Il, or his father). It seemed more than a little out of place, both because it really had nothing to do with the leader, and also considering he was talking to some South Korean family members. Also dare I say, it kind of reminded me to a small extent of how some people thank God. Not so much just as an expression, said without meaning or any real thought towards God or religion, but more like saying grace before a meal.
So yeah, things like that lead me to believe the mourning is quite genuine.
As for his son, I think I'm actually quite hopeful. By pretty much all accounts, he's quite unknown, so who knows what he'll be like. But honestly, considering the situation, I kind of feel like things only really have one way to go. So I'm pretty hopeful. I'm not really expecting to see any miracles or anything, but I'm sort of hoping for some gradual improvement for the region.