It's hard for people to make any convincing moral argument that doesn't eventually devolve into categorical reasoning ('this is right because it just is'), let alone on a deeply nuanced subject such as suicide. All arguments rely on a certain set of axioms or presuppositions that must be taken for granted in order to have logical grounds to stand on, but no idea in this world is ever truly self-justifying or fundamentally true. It is impossible to make a convincing argument for anything unless somebody understands a person's axioms very well and knows how to argue around them. No claim can be universally justified; everything is relative.
That said, this is a fun prompt, so I'm going to try to expound on this. I am not an authority on this matter. I welcome discourse if anyone wants to chime in.
To play the Devil's advocate and state my own view on it, what I personally like to consider are the ontological implications of the actionable objective of suicide: causing one's own death, resulting in annihilation of one's experienced self (assuming, of course, you do not believe in any afterlives or in spiritual permanence of the self).
Logically speaking, suicide is usually done with the goal of alleviating one's own suffering, or as an escape from the dissatisfaction experienced with life within one's own mental circumstances (or within society, or in more abstract cases even within the universe as a whole, etcetera). The supposition here is that one's self is the foundational cause of their suffering. In other words, by eliminating the origin or 'root' of one's feelings, the first principle from which everything else thereafter flows, it is a surefire way to ensure the end of your undesired experience and to be absolved from subjection to it.
A common rebuttal to this is that it also comes with the erasure of any possibility at improvement, or at experiencing any future form of happiness or pleasure ("it might get better!", "just keep trying!", etc.). From the suicidal person's point of view, this is often irrelevant, as it is much more preferable to him that he altogether avoids the X years of suffering that very likely are to be in store for him if he is to continue living, in comparison to continuing life in wait for the uncertain chance that some twist of fate may save him and cause a full recovery from his condition. I think most people would agree that if presented with the choice of enduring hardship with no benefit for it in exchange, or just preventing the hardship altogether, it would only make sense to choose the latter option.
Death, however, as far as the conscious self is concerned, is a form of prevention only in idea; in what is experienced firsthand by the subject, it is a form of erasure. To say that ending one's self is a preventative measure would be to make a category error. Even though one could say that future suffering can be avoided by a choice he makes now, avoidance requires you to be a conscious subject that is persisting through time and would have otherwise experienced the thing being avoided. By the death of one's self, they are erasing all possible emotional and cognitive states that could be experienced. You cannot logically compare "A state that I won't experience" (that is, if you kill yourself) vis a vis "a state I will experience" (that is, if you choose to live) if the first half of this comparison doesn't actually exist; your preference, after all, only exists so long as you are present. The suicidal, the "I" in this case, is evaluating the outcomes as if his preference will survive into the future, which, if he goes through with suicide, it won't.
By consequence, the act of suicide done with the goal of liberation from suffering is self-refuting in purpose. You are annihilating the very thing that you're seeking to benefit: yourself (even if it is through, say, removing yourself from the world so as to not be a burden on others; this still ultimately counts as a form of self-concern through seeking to achieve your exoneration from a perceived moral guilt). Suicide makes total sense only up until the very moment that you succeed to do it, and by then, you have already killed yourself too late. Under this view, life is defended not because any desirable things will happen if you live; they probably won't. Even so, being alive is the only condition under which any prudential value can exist for you at all.
This does not really mean suicide is a negative act, though. I view death as the most neutral, value-free state there is. It doesn't cause any suffering for the one taken by it, and it doesn't cause them any pleasure, and after all, we're all gonna die in the end anyway, so it doesn't truly matter when you die.
This argument is from a purely philosophical point of view, an argument "for the sake of the argument" so to speak. In reality, humans do not live like this. We are emotional creatures first and foremost, and the heart of man is a deceitful yet powerful thing. The best argument against suicide, in my opinion, often tends to be to not bother arguing against it at all, as it stems from emotion; rather, it is best fought with more emotion. One should strive to replace the suicidal person's misery and suffering with pleasure and joy, and then they will continue life regardless of how illogical or difficult it may be to do so. I have never successfully talked anyone out of suicide by providing them with logical counterarguments, but I've done it plenty of times by telling them I wish them well and that I'll be deeply miserable once they're gone. I myself even postponed my death because of a simple plea from my loved one.
Exactly 1000 words. Thanks for reading.