I hesitate to get into the abortion debate because it so often becomes sloganeering, but your discussion with Hippo has been thoughtful so here's my two cents worth.
The Supreme Court has basically got it right, in my opinion. The more advanced the pregnancy, the more interest the state has in the fetus, the more regulation the state can impose.
It's important to recognize how much this debate is influenced by advances in technology. For eons, personhood began at birth and for the most part it still does. Your birthday isn't the day you were conceived, it's the day you were born. Virtually every legal right you have is driven by counting your age from that date: the right to vote, to marry, to drive, to drink, to serve in the armed forces, to draw social security or qualify for medicare, etc etc.
Only very recently have we been able to "see" the developing fetus in utero and only those visions have led us to confer on it a kind of personhood before birth. When you add great advances in saving the lives of premature infants, you increase our sense of queasiness at terminating a pregnancy when the fetus might have been saved, albeit with heroic and hideously expensive interventions.
So when does a fetus become a person? In my view, the answer is when it has a reasonable chance of survival outside the womb. That's not the same as saying it should have ALL the rights of a living and breathing human infant, but that it is entitled to some protection.
Back to the Supreme Court and its trimester distinctions. Trimesters are arbitrary but the underlying logic is sound, in my view. In the earliest stages of pregnancy, when the vast majority of abortions occur, the aborted material is a ball of cells about the size of the period at the end of this sentence. I don't find that abortion morally troubling, though I realize that others do. By the third trimester, we are dealing with a fetus that is potentially viable as a live birth, even though it may require a lot of post-birth medical intervention to keep it alive until it can live on its own. At that point, the state has the right to put pretty significant restrictions on what can be done to that fetus.
That's the way it looks to me. Hope this was a useful comment.