Introduction
Libertarians come with all different moral values. Abortion is no exception to this. There are Libertarians that are pro-life and there and Libertarians that are pro-choice. But one thing Libertarians can agree on is that the government shouldn't be involved in the legality of abortion.
Life, Liberty, and Property
Libertarians believe that each individual has a right to life, liberty, and private property. As long as they don't impede on these same rights of others. They believe that government should only intervene if these three basic rights are threatened. Laws such as rape, murder, theft, etc. would still apply in a Libertarian society, but victimless crimes such as drug possession and prostitution would not. We'll apply these concepts to abortion later on.
The Pro-Life Aspect
Libertarians that believe in pro-life see the unborn child as a human life. Thus, since the child has life, the government should intervene to prevent the child from losing their life by the abortion. That, although the child is growing within a woman's body, they are a separate human being that deserves the same protections as any other individual.
The Pro-Choice Aspect
Libertarians that are pro-choice see abortion as the woman's option. They believe that any government intervention to stop the abortion from taking place would be impeding on the woman's liberty to make such a choice on her own. They don't see the unborn child as having the same rights as other human beings.
The Dynamic View
In addition to pro-life and pro-choice, there are individuals whom will possess a different view depending on the development stages of the unborn child. Some may lean pro-choice in the early stages of pregnancy and lean pro-life towards the later months. Of course, this would vary greatly between individuals in regards to when they switch from the pro-choice to the pro-life aspect.
The Potential Solution
Although, Libertarianism's primary goal is self-governance, I believe that government, at the state level, should enact law specifying at what stage an unborn child has the three basic rights of life, liberty, and property. Since this will be performed at the state level, laws will vary amongst the 50 states. Leaving individuals with options to live or travel to an area that would correspond to their view of abortion.
Considering the above, government will not have a say in whether or not abortion is legal or illegal within their state. Doctors and other medical professionals will still have a say as to whether or not to perform abortions. Even if the woman has requested the abortion before the unborn child would be recognized with having a right to life in the respective state. This allows for the individuals to resolve the abortion disagreement without government intervention and allows doctors to not be forced into performing an abortion that they, themselves, don't agree with.
A woman would have access to abortion under the conditions that she is within a state that doesn't recognize the child as having a right to life at the current stage of development AND is able to arrange with a doctor whom is willing to perform the abortion. In addition, the child will have the protections to life recognized by the state AND the individuals.
Finally, as an added benefit, the inconveniences and concerns of locating a doctor willing to perform abortions will hopefully promote safer sex practices. Which should cut down on the number of abortions being used primarily as a method of birth control.