When is a child granted the right to live with one single parent instead of visitation rights

It's actually very rare that a parent is denied any parenting time at all, unless there is some proven history of abuse by that parent, or some other good reason to believe that the parenting time would be against the child's best interests.

New Jersey Family Part judges will listen to children's wishes, if they are mature enough, but those wishes are far from the only thing that gets considered.  It is true that the older the child is, and the more intelligent and mature, the more the child's wishes will count for.

Typically,a minor child cannot choose which parent they want to live with.  The court will look to many factors - home environment, income and education of the parents, the child's relationship with each parent, reason for wanting the change, etc. 

However, while a child cannot outright choose where they want to live, if they over the age of 12, a child's preference is given consideration and is one of the factors that the court considers (the older the child, the more weight it is given).