As a Christian I find it rarely makes sense to try to convince a non-believer of some religious, spiritual principle, if they do not share the same foundation for establishing truth. However, when it comes to extremely important issues, issues that effect life and death, issues that effect public policy and legislation, which in turn affect us all, Christians understandably want to reach non-believers with their message. When this is the case Christians need to resist the urge to say that 'The Bible says ____', because, as it relates to public policy, and as it relates to convincing non-believers, what it says is not going to be relevant from their perspective.
Thankfully God has made many truths accessible even to those who do not believe. The most obvious way is through authoring the human conscience. However some issues are murky even to those of us who possess a conscience. Our conscience can tell us that ending a human life is wrong, for example, but our conscience can't tell us whether or not a just-conceived embryo constitutes a 'human life'. Luckily, modern science (i.e. embryology) can.
Since the pro-choice position is more prevalent among non-religious people, one would expect that the argument against abortion to be at least partially grounded in supernatural, religious presupposition and dogma. This turns out to not be the case. What we actually find is that the Christian argument against abortion is not contingent upon any specific scriptures (though it is often supplemented by them), or commands from God, and is an argument that can be made completely independent of any belief in any particular God. Though the Christian argument is often made with reference to the soul, and to various scriptures suggesting the soul is present at conception, it is not necessary to posit a soul at any point in order to show that abortion is wrong.
You need only to assert that the wanton destruction of a unique, innocent, human life is wrong (a point virtually all secularists and humanists will grant you), and that a just-conceived embryo is all of these things (which is almost just as obvious, but will take a little more to convince them of). Which, per the definition of each adjective, it factually is.
Thanks to modern science we know that conception is the point at which we become 'unique' (that is, when our own DNA is formed). Just before conception the unfertilized egg and the sperm are still parts of the person they came from, and, as such, the destruction of either the unfertilized egg or the acting sperm would not represent the destruction of a unique, individual entity. The secularist should grant 'unique' without any difficulty, I only mention this to preemptively counter slippery slope arguments such as 'well then is contraception/masturbation murder also?' No, they aren't, for the reasons stated.
'Innocent' should be no problem for the secularist to concede. That one is actually a more interesting debate among and between believers, rather than when dealing with non-believers.
'Human' should be fairly intuitive. The embryo has human DNA, not duck DNA, or platypus DNA, or pterodactyl DNA. This also should be granted readily, and if it isn't then whoever you are trying to convince is probably conflating 'Human' with the philosophical issue of 'personhood', which is not what this is about, because we are arguing the issue from a strictly materialist perspective. It's simply about which particular mammalian species we are discussing.
'Life' might be the only slight sticking point, though it simply is biologically, factually, just that. Biology recognizes single-celled organisms as life, and an embryo consists of many thousands (possibly millions? Biology experts help me out) times more than a single cell.
Thus, if you are against the wanton destruction of unique, innocent, human life, you must be against abortion.
Also, as an addendum, the secularist will often appeal to the social benefits of abortion in arguments that they make, however, even if there are many, I submit that IF we know that an embryo is a human life with rights, then whatever social ills (or benefits) that come from it existing are moot. There are many social benefits to be derived from euthanasia and genocide as well, but few would make the argument that those benefits make euthanasia or genocide justifiable.
Lastly the secularist will appeal to the rights of the mother. And in situations where the life of the mother is in danger, then this becomes relevant. Other than that though, your rights end where the rights of another begin, and the right to life trumps the right to choose.
I've always found it peculiar that the abortion issue is often fought along religious lines, or is at least perceived to be. Pro-life, anti-abortionists should be able to find allies in all walks of life, and across spiritual divides. I think this fact is illustrated well in Christopher Hitchens. He is of course a militant atheist, fervently anti- all things God, but he believes that abortion is wrong on materialist grounds (presumably arriving at that conclusion using some line of reasoning similar to what I have presented).