In "Respecting the Moral Agency of Women, Virginia Ramey Mollenkott says she can find no scripture to settle the question of "competing rights between mother and fetus," and thus settle the abortion issue.
This mindset reveals itself in statements by "pro-choice" advocates such as:
"An abortion when needed is a blessing. It is a gift, a grace, a mercy, a cause for gratitude, a new lease on life." --Valerie Tarico, Planned Parenthood Board of Advocates
“I feel like God wants me to do this job” --Diane Derzis, owner of the last abortion facility in Mississippi
Choosing abortion is "what religious liberty is about." --Ilyse Hogue, president of NARAL Pro-Choice America
[Added May 30, 2014 --
Why I perform abortions: A Christian obstetrician explains his choice
The Abortion Ministry of Dr. Willie Parker]
Aside from multiple references to the existence and personal characteristics of babies still in the womb (among them Genesis 16:11-12, 25:23; Luke 1:15,44), there is ample scriptural help for when we have to for decide between the competing rights of individuals. Look at what God says to slaves and masters, husbands and wives, Israelis and foreigners, rich and poor, etc. In short, the Bible is full of instructions concerning civil disputes in which one rival party is the "person" and the other generally considered "non-person."
God's word makes a strong case against treating anyone as a "non-person human being." Slaves are persons. Women are persons. Gentiles are persons. Poor people are persons. Scripture demands we treat all impartially, as persons made in God's image.
There are plenty of candidates for what is being called "non-person status" recorded in Scripture, but no allowance is granted for their mistreatment. The unborn child is, therefore, a worthy candidate for protection.
Related reading: In addition to scripture, I have found the article "Medical Ethics and What it Means to Be Human" by Donel O'Mathuna very helpful.
See also Scott Klusendorf's "Dead Silence: Must the Bible Say Abortion is Wrong Before We Can Know It's Wrong?"