Some thoughts that I don't remember anyone expressing on LW.
First let's get this out of the way: life does not "begin at birth". As far as life can be said to "begin" anywhen, it begins at conception. Moreover, the child's intellectual abilities, self-awareness or similar qualities don't undergo any abrupt change at birth. It's just an arbitrary moment in the child's development. So it would seem that allowing killing kids only before they're born is illogical. What are the odds that your threshold for "personhood" coincides so well with the moment of birth? Could it be okay to kill kids up to 2 years old, say? CronoDAS voices this opinion here.
But there's another argument in favor of considering the moment of birth "special". Eliezer linked to a study showing that the degree of parental grief over a child's death, when plotted against the child's age, follows the same curve as the child's reproductive potential plotted against age. Now, the reproductive potential of an unborn kid depends on its chance of survival, and the moment of birth is special in this respect. In the ancestral environment many kids used to die at birth. And mothers died often too, which made their kids less likely to survive. An unborn kid is a creature that hasn't yet passed this big and sharply defined hurdle, so we instinctively discount our sympathy for its reproductive potential by a large factor without knowing why.
How much this should influence our modern attitudes toward abortion, if at all, is another question entirely. As medicine becomes better, kids and mothers become more likely to survive. So if our attitudes were allowed to drift toward a new evolutionary equilibrium which took account of technology, we'd come to hate abortions again (thx Morendil). But then again, the new evolutionary equilibrium is probably a very nasty system of values that no one in their right mind would embrace now (won't spell it out, use your imagination).
Ultimately your morality is up to you and the little voices in your head. You think womens' rights trump kids' rights or the other way round, okay. But if you use factual arguments, try to make sure they are correct.
ETA: see DanArmak's and Sniffnoy's comments for simpler explanations. Taken together, they sound more convincing to me than my own idea.
This is a very important point, and I'd like to add to it. (I will use the term "ex vivo" to denote separation from the mother's womb, in order to avoid any issues regarding the term "born.")
It's worth considering if those who oppose abortion as murder actually believe what they say they believe. If the killing of a fetus by another being is murder as we tend to think of murder (when it comes to an ex vivo human), any fetal death that occurs otherwise (from disease, injury, etc.) ought to be just as objectionable as the death of an ex vivo human.
It is often seen that, when a miscarriage happens and is realized by the parent(s), a mourning process does occur. However, we do not simply mourn the deaths of born humans. We take active measures to prevent those deaths, through medicine, safety precautions, etc. Why don't those who oppose abortion as murder put as much, if not more, energy into preventing fetal deaths? There's prenatal care, vitamins, recommendations of things pregnant women should and shouldn't eat, drink and do... but compared to how much modern medicine we have to prevent the deaths of born humans, it doesn't seem like that much. Maybe it's just that abortions constitute a much larger proportion of fetal deaths, so they are the appropriate area of focus.
Nope. The majority of fetal deaths occur through miscarriage. One study found: "61.9% of conceptuses will be lost prior to 12 weeks. Most of these losses (91.7%) occur subclinically, without the knowledge of the mother." It's been estimated (source) that 50% of all pregnancies end in miscarriage; the 25% number is actually just for known pregnancies.
If those who oppose abortion as murder actually did so due to a deeply held belief, other forms of fetal death would get much more attention than abortion, because they occur much more often (a fact which is not hard to find). Even those measures we do take to improve fetal health generally aren't done for the well-being of the fetus in its own right; we take these measures to improve the potential ex vivo human's future. Likelihood to survive does seem to be the implicit criterion, with the understanding that becoming an ex vivo human, i.e. an independent being without a direct, physical connection to another being, is the criterion for personhood.
By the same sort of reasoning, we can conclude that people who devote more attention to stopping murderers (in the conventional legal sense: people who stab other people with knives, for example) than they do to stopping deaths by natural causes (such as old age, for example, which results in many many more deaths than murder) don't really do so "due to a deeply held belief" either.
Of course, another possibility in both cases is that we treat events as more worthy of attention when we think they are deliberate and intentional acts than when we don't.