The heir, the spare, and the priest

[Poliut]

So what do you have to do to get one of your sons named bishop? The last time I tried this, the heir/successor to my (arch)bishoprics bounced around from courtier to courtier for a bit before settling in on a son I wasn't intending to be a bishop.

[Elizardbirth the Sequined]

If you have Free Investiture as a Catholic you should be able to appoint a successor to a bishopric in the Religion tab. If you have Papal Investiture you will have to ask the Pope.

https://ck2.paradoxwikis.com/Crown_laws#Investiture

I think Fraticelli is the only other religion that allows you to appoint successors to bishops, everyone else gets a randomly generated character or a random heir from the bishop's court.

[EllyBeeShaped]

You can use the Pagan reformation option to include a religion style that lets you force children to take vows, too.

[al_x_ator2411]

I think high level theocracies are a bit peculiar in the sense that they take their own courtiers into account before generating a random new heir. But I wasn't aware that it interferes with Free or Papal investiture. :/

Try appointing him heir to a baron-level bishop and see if that sticks. You could also force him to take the vows.