If by "a mind of their own" you mean that brains are autonomous, and have thoughts of their own, independently of their "owners", that's true. We have as little control over our thoughts as we have over our dreams. Plain evidence (and a lot of renowned psychology authors, for example Jung) tells us so.
That's why, for example, Buddhists teach techniques to detach yourself from your thoughts, to learn to see them for what they are, something different from yourself, and thus avoid a lot of useless suffering.
Regarding brain growth, the theory for a long time was that neurons, i.e. the brain cells, were the only ones in human body that didn't reproduce: you had a set amount at birth, and as they died, there was no way around it, mental decay was unavoidable. However, now it has been discovered that the brain can create new neurons, and also, that how well interconnected those neurons are among them is way more important than the actual number of them; so our brains keep growing during our whole life, there's not a stopping point as it was thought before.
You have to take in consideration that human mind is the great mystery, perhaps nature's most sophisticated creation. There's a lot we still don't know. Scientists have analyzed the brain in search of where is consciousness located, and have found nothing... They distinguish all the "machinery" in there, but nobody has been able to pinpoint that "spark" that makes us be conscious, aware of things, in the way we are.