Abortion is probably the only philosophically reasonable difference between democrats and republicans.
The difference is, I think, entirely in the definition of "human". There is agreement that not everything which is alive and which has human genes, such as cell cultures and tumors, is a human being.
I believe that there is a non-discrete but important gradient between the not-yet-human and the fully-human. Some feel that there must be a sharp dividing line, which is reasonable insomuch as subjective definitions leave uncomfortable room for later injustices. The place they often choose for this line is conception, which certainly is a momentous event and the first moment at which a specific human genome will come into existence.
The severity of this philosophical divide should not be understated. Consider those who bomb abortion clinics. While if one considers full humanity an accumulated property, rather than an inherent one, the act of violence against provisors of abortion is abhorrent. In the opposite case, it is no less reasonable than violence against someone attempting to execute random adults.
Considering this, the best way to convince someone of your position about abortion is to convince them to share your definition of humanity. Compare this to the arguments generally made for or against abortion. For, the argument is that it is the pregnant woman's choice as to what they define life to be (which, while noble-sounding, is contrary to most law's objective definitions). Against, the arguments are usually biblical, which is insufficient for the majority of Americans and contrary to the constitution.
I would very much like to ask each candidate exactly how they define a human being.