Good question. We typically don't think of young kids as having neuroticism so much as an aspect of temperament we call "negative emotionality." Some children seem to have more of it than others. Taken at a slice in time, roughly half of the between-person personality variation on a given trait can be attributed to genes, the other half is environmental. (Strictly speaking, this doesn't quite account for variation in the change of traits over time though.)
Also, there's a well-studied effect where adolescents typically experience decreases in neuroticism / negative emotionality as they change and mature as they move into young adulthood.
Also, there's a well-studied effect where adolescents typically experience decreases in neuroticism / negative emotionality as they change and mature as they move into young adulthood.