I know of two ways in which people bring the idea of the soul into arguments about abortion. Note that in both cases the soul is mentioned in order to argue in favor of the permissibility of abortion.
First, some (usually non-religious) people dismiss the pro-life position from public consideration by claiming that it rests on the "religious" view that an unborn human being has a soul. But no one dismisses my opposition to murdering teenagers on the grounds that it rests on a "religious" view that teenagers have souls. Pro-lifers oppose killing the unborn for the very same reason they oppose killing teenagers -- because the victims are human beings, and it is wrong to kill innocent human beings. No appeal is made to souls or religion (although I do not believe that such an approach is automatically illegitimate).
The nature of the human person does play a crucial role in determining the ethics of abortion -- it plays that same role in determining the ethics of any kind of treatment of human beings -- but all sides of the abortion debate necessarily reflect a particular metaphysical perspective. Arbitrarily ruling one side out of bounds is not legitimate.
Second, some (usually religious) people claim that the unborn does not yet have a soul and therefore abortion is permissible. Such thinking seems to have its roots in biologically antiquated (literally) embryology. But it is hard to make sense of this view today. Presumably the proponent believes that at some point in her continuous growth and development (when, exactly, and why?) a human organism becomes "ensouled," which is presumably a supernatural act of God. Some living members of our species do not yet have souls (and some perhaps no longer have souls?), and we may kill them. Anyone who advocates this view should provide reasons to think it is true. I don't know of any.
Especially for those "ensoulment" theorists who are Christians of some kind, this view is bizarre and impossible to justify biblically. Nothing at all in the Bible suggests that there could be such things as "soulless" human beings whom we may therefore kill. On the contrary, the Bible says God made man (human beings) in His own image -- not some men, but all men. This is affirmed by many passages that imply that I am identical to my prenatal self, and that God cares about human beings in the womb. If my soul is my immaterial self -- me -- then biblically that soul has existed from conception, because the Bible implies that I have existed from conception (e.g., Psalm 51:5, Luke 1:41-44, Psalm 22:10, Job 31:15).
(Apparently the Catholic Church is agnostic on the precise moment of "ensoulment," but holds that it is irrelevant to the ethics of killing an unborn human being, which is a serious moral wrong.)