The first element involved in Changing or reconstructing a  system is to understand its dynamic. Family Court, first and foremost, is equity court. Formerly known, with some deviation between countries, as a Domestic Relations court; Family Court is established to determine "who" will have child equity. 

So, what is a Court of Equity?

That is a "good" question. And, the answer is somewhat vague:

A court of equity is a court which can apply equitable remedies to disputes. These courts operate within the legal system, but rather than focusing on the application of law, they look at cases and determine outcomes based on fairness. Also known as chancery courts, they can be found in many regions of the world. In some areas, they operate entirely separately from courts of law, and in others, a court of law is empowered to handle both legal and equitable remedies.

The concept of a chancery court arose in England when citizens began to express dissatisfaction with legal judgments handed down by the courts. They argued that the law was sometimes unfair, and that some situations were not covered by the law, making it impossible for the courts to respond. Courts of equity arose to handle legal situations in which people might want damages beyond monetary damages, with the judge empowered to act on discretion, rather than following the rule of the law.

A court of equity still has some legal responsibilities, but it has more leeway when it comes to judging cases. It can hand down a judgment which includes an equitable remedy such as an injunction, as opposed to simple monetary damages. These courts can be used for things like specific performance, for example, in which someone is asked to make good on a breach of contract. 

The judge in a court of equity can weigh many different sides to a case and explore different perspectives to arrive at a judgment. This gives the judge an opportunity to use discretion and judgment, rather than having to rely on the narrow confines of the law. The outcome may be deemed more fair or equitable in those cases heard in a court of equity.

In areas where courts of equity and courts of law are separate, people may have the option of filing a case with one or the other, depending on the type of case and the policies. In areas where courts are empowered to act in both capacities, the judge can consider the issue of equity when developing a sentence or responding to a request for an equitable remedy. Judges who serve in such courts must demonstrate a capacity for critical thinking, discretion, and fairness beyond a basic understanding of the law, as the judge's capacity for evaluating a case fairly can be critical in a court of equity.

Why doesn't it seem to have anything to do with child custody?

Another great question - the closest reason is that at the onset of Equity Court - closely aligned to Estate Matters, at the time the law was created, children were considered Estate Matters.

In the best of all worlds, the judge ruling in a Court of Equity is fair and not swayed by associations, politics, revenue, status, etc. However, in the real world, this is no longer true.