Most obviously: it is unfortunate for anyone to be in the position of wanting or needing an abortion. It would be better if there were fewer unwanted pregnancies to begin with. Surely everyone can agree with that. So improved sex education and access to contraceptives ought to be supported by both sides of the abortion controversy. (All the evidence suggests that abstinence-only approaches simply do not work.) As Jonathan notes, this is no 'compromise' on the part of either party - it's a genuine common ground where they should be entirely in agreement. It's a disgrace to both sides when their partisans fail to recognise this.
Secondly, the question of whether abortion should be illegal is a separate matter from whether it is immoral. Some conservatives have fundamentally misunderstood the relation between law and morality - they do not recognise that sometimes immoral things should nevertheless be legal. But it should be obvious that this is sometimes true - consider lying, breaking promises, adultery, or (to a Christian) worshipping false idols. So the only question is whether abortion, too, is such a case. I think it is. Outlawing abortion would not stop it from occuring. It would merely drive the practice 'underground', making it far more dangerous. To outlaw abortion would put women's lives at risk. Even those with moral qualms about abortion should not be willing to pay the hefty social costs of such legislation (or, at the very least, it should give them pause).
More controversially, I think the reasonable person should agree that conception is not a metaphysically magical moment. The idea that 'life begins at conception' is contradicted by biology (a point powerfully made by Prof. Lord Robert Winston in a lecture I heard last year - I only wish I could remember it better). Scientists have created healthy mice that were never 'conceived' as such - the egg was artificially stimulated into replication and growth despite not being fertilized (I don't recall the exact details - possibly two of the mother's eggs were fused together to get the chromosome count right). And what to say of identical twins? Two persons formed from a single zygote - you do the math.
Perhaps most telling of all is the sheer fact that "between 60 and 80 percent of all naturally conceived embryos are simply flushed out in women's normal menstrual flows unnoticed". How does the proponent of life-at-conception deal with such a fact? Or, as Jason asks, "Why aren't they doing something about the 60-80% of all innocent human lives that nature spontaneously terminates? This stuff makes the tsunami look like nothing at all."
Single-celled zygotes are not persons, nor are they somehow "morally equivalent" to persons in any sense that entails a 'right to life'. To suggest otherwise would seem to entail that we have a moral duty to constantly monitor all sexually-active women, to save those "human lives" that would otherwise get flushed away without anyone ever knowing. But we have no such duty, and any suggestion to the contrary is absurd. Menstruation is not manslaughter. Those cells are no more 'persons' than are the cells in my fingernail clippings.
Pharyngula offers some helpful diagrams to illustrate this point, and comments:
Figure 1. These are children. Figure 2. These are embryos. I can tell the difference between 1 and 2. Why can’t you?
When you tell me you think an embryo is the same as my kids, you cheapen the worth of my children. They are much, much more than that small thoughtless blob. You reduce the value of family to mindless chemistry and metabolism.
Lastly, there is everything David Velleman says, from which I'll quote a few key points:
When we ask whether the fetus has a right to life, we make it sound as if there is a single thing, life, to which some beings have a right and others do not, or to which some beings have a stronger right than others. Yet what gives persons a stronger right to life than penguins, and penguins a stronger right to life than petunias, is precisely that these beings live very different kinds of life: the life of a person is a different process from the life of a penguin or a petunia. The reason why different creatures have different rights to life is that some kinds of life-process require stronger justification to end than others and result in greater wrongs when ended without justification So there is no one thing, life, to which different creatures have different rights; rather, there are different kinds of life, with different entitlements to continue.
The belief that a human conceptus is not yet a person in the early stages of gestation rests on the fact that it is not yet capable of the beginnings that make a person's life wrong to end. It is incapable of those beginnings because it does not yet have any mental life.
Here are some of the relevant developmental milestones.1 Synapses do not begin to form in the cerebral cortex until the 12th week after conception, and neurons continue migrating into the cortex until the 20th week, which is the first point at which electroencephalographic activity appears. The fetus's EEG doesn't coalesce into "waves" until in week 26, or develop the patterns characteristic of waking and sleeping until week 30. Synapses begin to connect the spine to the thalamus in the 20th week and reach the cerebral cortex between weeks 24 and 26; not until the 29th week do peripheral stimuli evoke measurable potentials in the cortex, indicating the completion of functional sensory pathways.
I leave open the question of when killing a human organism becomes morally impermissible. The neurological facts suggest, I think, that setting the boundary any time before the 20th-30th week would be quite unreasonable. But that still leaves a lot of room for disagreement over issues like late-term abortion (though I'm not even sure that newborns are persons in the relevant sense).
To be fair, really only the first point discussed here would count as geniune 'common ground'. But I do think the other claims I've argued for here are well-justified. One might reasonably argue that fully-developed foetuses should be considered persons. But it is not reasonable to say the same of a single-celled zygote.