An orthodox Catholic I know cares more about abortion than any other political issue. He votes for candidates based largely on his expectations about the kinds of judges they'll appoint or confirm, behavior I completely understand given the certainty he feels that every abortion is a murder. At the other extreme are pro-choice voters whose number one issue is protecting Roe vs. Wade from being overturned, preventing any restrictions on abortion, etc.
These are by their nature long term political struggles, or so you might think: the composition of the Supreme Court is always going to change, legislatures can be influenced to hue closer to one side or the other, etc.
But I predict that what we now think of as the abortion debate is going to radically change within our lifetime in a way that makes many of the strategic gambits employed by both sides irrelevant, or at least beside the point.
Specifically, I think that technology is going to make fetuses viable outside the womb earlier and earlier. In fact that is already happening. And eventually there will be artificial wombs, enabling doctors to extract a fetus from a pregnant woman during the first trimester with a procedure no more invasive or dangerous than abortion, and to keep that baby alive in an incubator.
Today we are used to thinking about a woman's right to end a pregnancy as the functional equivalent of ending the fetuses' life. In the future, however, that need not be so. A woman could be afforded the right to end her pregnancy, but be denied the right to end the life of the fetus. Although I am not an expert in abortion jurisprudence, it is at least conceivable that this could happen without any need to overturn Roe vs. Wade.
It is conceivable that adoptive parents would step in to raise children who would've been aborted prior to artificial womb technology, though it is unlikely that enough adoptive parents could be found to raise all the children now aborted. It is possible that society's views about killing fetuses would change in the pro-life direction once that change didn't entail forcing women to carry an unwanted pregnancy to term, and that the government would be in the business of funding large scale orphanages.
It is even conceivable that women would find themselves in the same position that some men find themselves in now: forced to pay for the upbringing of a child they'd rather have aborted. Were I a strategist at a pro-life or pro-choice advocacy group I'd be spending a lot of time and effort figuring out when changes like these are going to happen, what I thought about them, and how I could shape them to advantage my side.
UPDATE: This paper considers similar arguments in far greater detail.