They need to be in charge of armies, or else the heirs don't get any vnv's (IIRC). And as Vlad suggested, the best way is to take more territory. That way they get command vnv's and stars, and I also think that influence is directly tied to command stars (right, vlad?).

As far as crappy heirs go, if I can't build them up, or I get a terrible heir right in line before a very good one, I send the chinless wonder on a one way trip to Mongol territory (or in your case, whomever you're at war with). You can select for good leaders that way, though it does waste a good unit of heavy cav.