Children's "rights"* is a topic of confusion and uncertainty among libertarians. I'd like to solve this once and for all, but not right now. Perhaps someday. Today, I only ask libertarians to consider the logical conclusions of either 1) children owning themselves and therefore possessing all the "rights" associated with ownership or 2) children are owned by someone else, probably their parents who created** them.
With 1) we must immediately start applying the non-aggression principle (NAP) to our interactions with children. If they are self-owners, then we have no right to hurt or handle them against their will. Will can certainly be tricky with a being that can't quite communicate (infants), but the NAP demands we ere on the side of non-aggression. Nonviolent, non-punitive, peaceful parenting as well as radical unschooling are the logical conclusions of how we should raise our children. Anything compulsory or punitive would violate the NAP, and our children's "rights" and autonomy.
With 2) our children are slaves. Parents (or whoever) have it within their rights as owner to do whatever they want to their own property. They can handle it softly, or violently. If we accept 2) we must be prepared to accept that parents can not only give love, hugs, and kisses, but also hit, beat, abuse, rape, trade, and kill their children.
The only other alternative is some sort of joint-ownership between children (of themselves) and their parents (or whoever). But even a joint-owner can't do as they please with their property without the consent of the other owner(s). A child cannot give meaningful consent to what would amount to aggression (they always accept non-aggressive acts because if they accept it, they consider it non-aggressive), and if they could, 1) may still apply.
Food for thought.
* I put rights in quotations because its positive shorthand for a negative concept. See this post.
** Is there a part of the child that is not created by the parents? A soul, or spirit of sorts?
Update 2/27/13: Lots and lots of discussion on this here.
Update 3/1/13: After all of the discussion I've had on this the last few days, it seems the above is incomplete. I will add this, from Debbie Harbeson, "If one accepts that the principle of non-aggression is valid, there is no need to put children in a separate category of discussion at all is there? If you believe in non-aggression then it applies universally." Very true. I guess each of ask must ask ourselves why it is we follow the NAP. Is it because we want to treat others peacefully and respectfully, how we want to be treated, or is it because we fear retaliation? In other words, would we violate it if we could get away with it? If the latter, perhaps that's why you feel it's okay to hurt or coerce children. I think the NAP is even a good approach to animals. We've all got to eat, but let's not waste life needlessly or treat it cruelly.