Please note that this is not the official position of the Libertarian Party of Oregon. This is an example of thought as viewed through the lens of liberty and freedom.
Libertarianism holds no official position on the issue of abortion. A Libertarian applies principles to formulate a personal opinion on any number of issues. On the issue of abortion the majority of Libertarians would be against abortion based on the application of the non-aggression principle. Then there may be Libertarians who side with choice based on private property rights, in other words, your body is your most sacred private property and trespassers maybe removed. The one caveat is that Libertarians also recognize there can be extraordinary situations that need to be evaluated on a case by case basis by the individuals involved and that a one size fits all solution is no solution at all.
Libertarians do agree that society should be organic and not engineered by central planners that are far removed from the families and communities that their interventions interfere with. Libertarians believe that the use of force by government to enforce a pro choice position or pro life position is not acceptable. That a culture should grow through dialogue and persuasion, community mores, and family values. This can be achieved by removing government incentives that has made the family unit financially and socially irrelevant to raising children in an economical sense. A strong family base with principled behaviors, modeled by the adults, is the natural way to raise children. These children then can mature into principled adults, who then value human life. In the end a society is best off living organically and not driving any behaviors underground with coercive laws. It is best to minimize certain human actions. However, trying to eliminate certain human actions only achieves negative and unintended consequences in the long run.