But we ascribe special significance to the time just before dying. It is seen as much more important to be happy the last hour you are alive, as opposed to any other in your life.

Why is that? The only reasonable explanation I can come up with is that it isn't really important for the person dying, but only for the rest of us. When we remember that person we will remember their last time especially strongly, and it is nice if that is a good memory.

Or hmm. Another view that would justify it is that life is about optimizing some function. And that before we die we only have estimates of how well we are doing, so it is only just before death we get the final answer, if we succeeded in optimizing the function or not.