Interesting question. I have not read much material on Rothbard’s ethical system, so I can’t speak for him, but here are my thoughts:
(Disclaimer to any trolls or shallow statists - no straw men of my argument just to ridicule me or show that libertarians are cruel, please)
First off, in a free society justice exists to make the victim whole. It does not exist, as it does right now, to emphasize some abstract collective right to justice that society has as a whole. That is, when a thief is brought to justice, it’s not done so that “society” is safer - it’s done because the victim has his TV stolen (or what have you).
As such, it would (at least initially) appear that a mother killing her child cannot be sued by anyone in society. If she did get sued, it would in no way go toward righting the abhorrent wrong she committed. What purpose would it serve? It would make society feel better about itself in punishing an evil person, sure - but that’s not the purpose of justice. That’s just called society feeling good about itself.
Now, here is where it gets interesting, and in fact so interesting that I’m tempted to write an article about it:
In a free society, the mechanism of justice is in fact a subset of larger, peaceful social mechanisms.
That’s a confusing sentence, so let me explain what I mean with an example:
In a free society (in my view) we would not have prisons like we have them today. People wouldn’t be “locked up.” Instead, criminals would (after they pay restitution through a court of law) be shunned by society. Who wants a thief on their land or in their store? It would come to such a point that the criminal is so cut off from the division of labor and fruits of society that he would have almost nowhere to go and nothing to do. That’s when the entrepreneur steps in and creates free market prisons. These would not be there to lock criminals up, but to in fact serve as a sort of hotel for them. The prisons accept prisoners into themselves and give them a safe place to live and to work (given that no one else wants them). In this sense, prison is a place where a criminal chooses to go rather than a place where he is forced to go.
How does this tie into the larger theme of my post? Well, notice that in “bringing the criminal to justice” (even in the socially feel-good way) all the things society did to him it could have done to anyone else legally! That is, the mechanisms by which society punishes criminals are not categorically different from other actions society may engage in (I use the words “societal action” loosely - society obviously doesn’t act or think as a separate entity, but only as an aggregate of individual actions).
In this sense, society can apply the same exact punishments to two very different people:
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A criminal who actually violated someone’s property
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A peaceful person who is being a total and completely-unbearable jerk
Going back to the mother example, though no one has legal standing to sue in court (because the victim cannot be made whole - besides maybe the other parent asking for some sort of restitution, idk) - again, though no one can sue, society can apply the same exact punishment to her as it could if there were a person with legal standing to sue. The only thing missing is the restitution to the victim - and only because the victim is no longer there to accept it.
To wrap it up, the interesting thing about a libertarian society is that it can punish abominable behavior in voluntary ways - such as the mother murdering her child.
I know that it’s a difficult concept to wrap your head around, but I hope you’re convinced that it ultimately makes sense.