I would really like some more insight on this.

Thank you Fishpants for posting!

"It's not (entirely) random. Try having a captain win a significant battle by a significant margin, and you should be offered the chance to adopt him into the family."

The only captain I have ever had promoted to general was one guy, with a unit of equites, running down a unit of rebel velites.

Meanwhile, my captains who have won maybe a dozen battles fighting massive gallic stacks, leaving crossed swords battles on the map even, have not been adopted.

Of course, my *regular* family is busily not producing any children ever. I've had males with the 'prolific' and 'fruitful' traits marry early and then their whole lives never have one child.