Abortion debates exist in moral forms and legal forms. When discussing the matter, it is important to decide which is being discussed and stay on category.
If legality is being discussed, I think the most logical place to start is with the determination of personhood. When does personhood begin? For the ones arguing for anything before birth, the challenge will be to explain why this isn’t applied across the board.
Building on @Bradskii’s example, there are “natural” abortions that happen all of the time. When a fertilized egg fails to attach to the uterus wall, it dies. Scientifically, sure, this was individually a new organism. However, no one seems to be bothered when this happens, science (to my knowledge) is not trying to find methods to save these individuals, it is just thought of as part of the process. No funerals are held for these individuals. They are not legally persons.
When an egg does attach and further develops, these individuals are not legally “persons.” They are not claimed on a tax return, they are not issued a Soc Security number, etc. According to the State, they do not exist.
It would be difficult to categorize an act as murder, when there is not a legally recognized “person” to be killed.
In my opinion, any attempt alter the legality of abortions would begin with changing the definition of personhood. The reason why birth works so well is that it creates a defined point in time to make the designation.