Please know that I'm not condoning suffering or evil, nor am I even equating the two in this OP (I think they're extrinsically relatable but not intrinsically relatable). I'm not saying the suffering of children is good, or that natural disasters serve some sort of intrinsic function.
1) Any good that comes from suffering or evil is extrinsic to suffering or evil. That is, evil or suffering in themselves are bad, but they can be turned into good depending on the perspective that frames them during and after the instances of evil or suffering. "For those who love God all things work together for good..." (Romans 8:28, ESV). Note the "for those who love God" part. This doesn't meant that people who don't know God can't get a better perspective of things after the fact and grow by experiencing an evil or instance of suffering that's bad in itself, just that there's something about an authentic (i.e., not just conceptual) relationship with God that involves getting the full deal of advantages from these bad things. What this also means, theologically speaking, is that God can hate evil but still use its inevitability to help his creatures. Another point is that evil can be understood as a negation (the negation of good), and our natural sinful state is a negation as well (the negation of how we should be in terms of having faith and being in a continuous, full relationship with God); from this, we can understand the "good" that comes from evil as the negation of a negation, rather than a "positive good" like when we see something beautiful. The "use" of evil in a good way is neither an intrinsic good nor a positive good, but a double negation equivalent to good, necessary only because we have sinful natures. "Why did God create people with sinful natures?" Because sin is inevitable with human freedom; you can't have a self without free will, and so God can't logically create selves without an inclination to sin.
2) Suffering can be useful in that it helps us value things we naturally don't value. I had a health breakdown five months ago. It was awful. But the biggest thing I took from it is an overwhelming appreciation for very basic things, and I think this appreciation is there in perfect proportion to how much I thought I wouldn't make it through this crisis. That is, to the degree that I thought I would lose everything, I gained a newfound appreciation for things. The big point is that without this sense of suffering, I would not have been capable of valuing these things -- sunsets, rainy weather, an angry wife, crazy coworkers, ad infinitum. What this means is that value is born from (a sense of) loss and the suffering and possible evil involved in loss. And without value, things are just things, scenery, taking up space.
3) Suffering results in "soul building," to use philosopher John Hick's phrase, in that only in a world of suffering is there the possibility of growth that would otherwise not take place without it. So without suffering, you can imagine a person being completely spoiled and unprepared for life given unrealistic expectations that things will be fine, which will obviously only set the person up for personality problems or even psychopathy if they don't get to experience and even embrace the suffering (and possibly overlapping evil) that's involved in life.
4) Suffering is a necessary result of human freedom, or free will. We know how this applies to most situations, but it also applies very importantly on a systemic level, which influences things we would otherwise think are just naturally there, such as children who grow up hungry or even natural disasters. We're responsible for the negative systems that influence these examples. With hungry children we're responsible because we're incredibly disproportionately greedy and materialistic, which takes away from our inclinations to altruism which would otherwise resolve this problem; and with natural disasters we've piled up economic externalities in terms of pollution which influence negative weather patterns leading to more natural disasters, and poor people are much more likely to be exposed to areas much more prone to natural disasters, like the poor in India who are much more susceptible to flooding.
5) From a theological perspective, you have the possibility of a judgment, which will punish those who misuse their freedom (i.e., everyone to varying degrees) which result in negative outcomes (suffering, pain, poverty, etc.) for different people (and animals?) in the world, while those who have suffered will be elevated over those who have inflicted suffering. From a purely logical standing, this makes a theistic world with evil that can't be explained better than an atheistic world where evil is just a brute fact, given that at least in a theistic world you have the possibility of reconciliation and restitution of all things, which you don't have in an atheistic universe.
IMHO, anyways.