The original Don Marquis paper "Why Abortion is Immoral".
Many moral theories agree that the bearers of rights should be “persons”, not merely biological humans. What are the counter-arguments for it?
I am not sure if this is the named argument, but there is a standard pro-life anti-euthanasia argument that brain dead people have recovered function, even if they did not awake with all their previous memories. So if we had let them die during the gap when they were not people, we would have been killing that person, who they were all along, despite the gap, because they were at that point potential people, as is proved by their recovery.
If the possibility they might recover means we should not kill them, even though at that point they have ceased to be actual people, then that extends in the abortion case to the possibility the fetus might be born, and you get the same result.
The standard counter-counter argument is that that previous person, before brain death, and this new person are not the same person. The notion of a personality requires a continuous experience of some kind. That means nobody would have been killed, the new person would just have been prevented.
(A better argument, to my mind, is that doctors make mistakes, and people die as a result all the time. The doctor can simply be wrong about that person having been brain dead, and there is no philosophical significance to human error or exceptionally tricky diagnosis. But it is a derailment to simply insist a philosophical argument is about nothing.)
So, given the link, this is not the same argument. But it works from a similar point. If having a potentially valuable future is what indicates you should not be killed, then that is the just about the same as potentially being a person in the future. It is slightly more abstract, so it reaches farther, which makes it even less compelling.
To me this has two flaws beyond the other argument:
1) Every sexual encounter could potentially result in a life. So this argument suggests birth control is murder. In fact, the logic that anything that might result in a future life is valuable, and that not allowing it to run to completion is killing, suggests that passing up any opportunity to have heterosexual sex is killing - (Cue Monty Python's "Every Sperm is Sacred"). There are potential people all over the place, and if we produced them all, we would have a big problem.
2) There is no indication that having a future is what makes your life valuable. Killing is resisted by an actual psychological fact -- it is naturally traumatic for people to consider ceasing to exist. There are times when this is passive, when we are sleeping, dissociated, psychotic or very high. And that is not an excuse for ending a life. But if something or someone persistently does not have that psychology, either because they are over it already, or because they have not yet developed it, it is ambiguous whether there is any real killing taking place.