A few things seem to contribute to the votes of your vassals. First thing I have noticed is that they will hardly ever support infants or young children. If you are in that situation, they will often vote for your sibling or another branch of your family. If this is the problem, they will start supporting him around age 10 or so; I'm positive he doesn't need to be a full adult.
Secondarily, both your personal honor and the honor of your son seems to influence this. I assume this is so because high honor is generating those positive modifiers to the relationships, and who they vote for is ultimately determined by vassals relationship to candidates. This is also probably why you are able to obtain court ranks for your eldest son. That, and the fact that it makes the honor hit at clan leader death less painful.