The argument from fetal potential usually is presented like this:
Premise 1: All persons have a moral right to life.
Premise 2: Since all persons have a moral right to life, all potential persons also have a moral right to life.
Premise 3: The human fetus is a potential person.
Conclusion: The human fetus has a moral right to life.
Usually, neither side disagrees with premise #1. Premise #3 is also largely uncontroversial, although one might argue that the reality of stem cell research involves using embryos that are left over "spares" from in-vitro fertilization, which would simply remain frozen or go to waste if not used for research. Anyhow, for obvious reasons this defense does not apply to the abortion debate, so we'll ignore it for now and take the argument from potential at face value.
The usual defense is an objection to the second premise of the argument ("people have a right to life, so the same must be true for potential people"). It has been pointed out that the argument from potential is never brought up outside the abortion and stem cell debates. To claim that if Z is true for X, it must also be true for a potential X is seen as logically lacking. Bioethicist Peter Singer, for example, points to cases where this is obviously not the case:
There is no rule that says that a potential X has the same value as an [actual] X, or has all the rights of an [actual] X. There are many examples that show just the contrary. Pulling out a sprouting acorn is not the same as cutting down a venerable oak. To drop a live chicken into a pot of boiling water would be much worse than doing the same to an egg. Prince Charles is the potential King of England, but he does not now have the rights of a king.
(Peter Singer, Practical Ethics. Cambridge University Press 1993, page 153)
More absurd examples are thinkable: As a physically healthy young male, I am a potential soldier, yet I do not have the rights or duties of one. I am also a potential husband and father, but that is hardly enough to make me eligible for the states family support. I am also physically able to rob old women, but can't be convicted for crimes unless I actually committed them.
Pointing out examples like this is usually enough to win the average pro-life-vs-pro-choice-debate. There are some people though, who are not that easily thwarted.
I've recently stumbled across a paper by Bertha A Manninen, published in "Philosophy, Ethics and Humanities in Medicine". (All following quotes are from this paper.) In it, she seeks to refute Singer and those who share his views, arguing that in many cases potential does, in fact, matter:
The task that the pro-potentialist now faces is to explain why the fetus' potential is a morally relevant characteristic that justifies extending to it a right to life. I believe that this can be done. What I want to do now is refute Paul Bassen's point that " [n]owhere outside the abortion debate itself is there a precedent for supposing that future prospects can create a present sake " by discussing examples that seems to run contrary to this assertion.
The first example she gives is the moral right to a health insurance, as demanded by article 25 of the United Nations' Universal Declaration of Human Rights.
[...] potentially ill individuals, like my child or myself, also have a moral right to health insurance. But why do we accord this latter group health insurance, even though they are not actually sick? Because possessing health insurance, even in the absence of an impending illness, also constitutes a great benefit for potentially sick people and a deprivation of health insurance also constitutes a harm for potentially sick people. (Emphasis in the original)
She goes on to cite the moral right to an education and some other examples, to finally arrive at the conclusion that
A potential X may be granted the same moral rights as an actual X in virtue of its potential if its potential generates an interest in such a moral right; that is, if possessing the moral right constitutes a benefit for the potential X and a denial of the moral right constitutes a harm. (Emphasis in the original)
Which, I think, is spot on. The only problem is that it does not at all relate to the case of a bunch of cells that we call an embryo, not even to an advanced human fetus. The key point here is the term of "interests", which a bunch of cells cannot be said to have. In all the cases Mrs. Manninen mentions there is indeed a benefit to the potential person, but not because of its potential.
Taking away a healthy person's health insurance, for example, does not constitute a harm because the person is a potentially sick person and thus has a potential need for health insurance. A healthy person has a very current interest in health insurance, because it provides a sense of security and thus quality of life. Having health insurance is a current interest, because it is the basis of current considerations and actions. It is the mental capacity of the human of perceiving himself as a person in time, of being able to think about the future and to acknowledge the fact of dying that makes the adult human exceptional in this regard.
The harm is not in taking away a health insurance that is not currently needed. The harm is in taking away the comfort and security such an insurance provides. Being able to perceive the future makes us able to feel discomfort, insecurity and fear at the prospect of having to face every day dangers without an insurance to lessen the impact of possible accidents.
Embryos, on the other hand, distinctly lack that property. An embryo cannot be said to have an interest in its future life, while grown humans can very well be said to have an interest in health and other aspects of their future.
Finally, the center of the argument from potential remains the equation of a tiny bit of biomass to a fully conscious intelligent being. This argument must fail because the extent of "harm" one can inflict on something is depending on that something's interest in avoiding that harm. Indeed, harm can be defined as the disregard of an interest.
Slapping a person's face , for example, is generally considered to be harmful, but only because most people have an interest in not being slapped. In a consensual situation between a sadist and a masochist, however, slapping might even be considered beneficial to both parties because interests are being served rather than disregarded. Much like a stone, an embryo cannot be harmed because it has no interests - not even in its own continued existence.