Abortion is a problem here and everywhere, now and forever, because it interferes with the creation of new life. Libertarians believe fiercely that individuals have rights. Before a woman gets pregnant, she is an individual. Shortly after she gives birth, there are two individuals. Somehow, a new individual was created. This individual's rights must be respected, without initiating coercion against the mother in whose womb the baby was created.
A standard principle of libertarianism is that the best solutions are discovered when people have the most control over their own lives. Given private property and free markets, people will negotiate and trade to improve their circumstances. A difficulty with applying this principle to abortion is that neither a zygote, a fetus, nor a baby are particularly at will to enter into these negotiations. There are enough people who have an interest in protecting a baby's rights that they can act as a reasonable proxy for the baby's interests.
The libertarian problem here is that the baby has, without any intention on its own part, found itself at risk of loss of life without cooperation from the owner of the womb it needs for its nurture. The baby has done nothing wrong. The mother may or may not have done anything "wrong". The mother may have been reckless and taken undue risks of accidental impregnation. The mother may have taken reasonable precautions. Or the mother may have been raped. What is clear is that the mother does not wish to cooperate, and history has proven that cooperation cannot be easily coerced.
Pregnancy is similar to other legal quandries. Let's say that a person needs to use the resources of another to save their life, and cannot negotiate the use of those resources. A reasonable law will let them use those resources, as long as they "make the owner whole". That is, they must restore the owner's property to its original condition, and compensate them for the use of their property.
I think, then, that a libertarian solution to abortion is to allow a mother to rent her uterus to the baby. On a practical basis, that is what many parents do. Parents expect that their children will take care of them in their old age, just as they took care of the children when they were helpless and feeble. The trouble comes when a mother doesn't want the baby. Of course, there are these days any number of parents who are unable to have their own child and are willing to expect resources to adopt a baby.
So, you have a willing buyer, and a willing seller. Why not sell babies? There are obvious moral implications, in that it's akin to slavery. Purchasing a baby is nothing like slavery, though. The slave purchaser expects to get a return on their money without much further investment. Purchasing a baby is more like buying a car that needs repairing. You know that you'll have to spend more before you'll get any value.
Or, rather than buy and sell babies, perhaps anti-abortion groups could act as baby brokers. They could take a payment from someone who wanted a baby, be responsible for the actions of that person, and use the payment to compensate someone who didn't want their baby and wanted to give it up.
This would work just fine if there were no unwilling sellers. That is, if every woman had a price for which she would allow her womb to be used, then it just becomes a matter of finding enough money to clear the market. Doubtless, some women would be unwilling to allow their womb to be used for someone else's nurturing. In this case, the whole problem comes down to eminent domain. Would it be possible to "take" a woman's womb for use by a baby (that is, for public purposes). Clearly, if there were enough willing sellers of "womb services", it would be possible to establish a fair market value, and compensate women for the use of their womb.
Basically, then, the failure of current and past abortion laws to make enough people happy comes down to the confiscation of private property for public purposes without due compensation. I believe that if abortion was illegal beyond a certain date in the pregnancy, AND a woman was fairly compensated, then you would see more people cooperating with such a law.