This is an argument that relies on a few definitional premises: thus we can also pre-empt a few other responses by noting that if the premises don't apply, then neither does this argument; e.g., if you believe in a God that doesn't meet the definition provided in the argument of omnipotence, or omnibenevolence, or whatever, then the argument does not apply, so there's no need to point that out.
What sort of God is this argument aimed at?
This argument is aimed at any God that's claimed to be:
Omnipotent: For the purposes of this argument, omnipotence is the capacity to actualize any logically possible state of affairs. This avoids silly paradoxes like "making a rock so large He can't lift it," or any other irresistible force meets immovable object type paradox (these are not logically possible, so we don't have to worry about them).
Omniscient: For the purposes of this argument, it's sufficient for an omniscient being to know every possible truth and avoid believing every possible falsity. We do not have to worry about deeper nuances such as "can God predict an exact free will choice," that will be irrelevant here.
Omnibenevolent: This is probably the trickiest one to nail down. For the purposes of this argument, a being is omnibenevolent if it doesn't desire to cause suffering and purposefully takes every measure logically possible to prevent its instantiation. This definition leaves open the door for certain theodicies which will be addressed piecemeal.
The argument proceeds like a traditional PoE: the existence of physical suffering is incompatible with the existence of an omnipotent, omniscient, and omnibenevolent creator. What is meant by physical suffering? I mean just that: any form of suffering ultimately caused by the physical universe, such as stubbed toes, disease, aging, animal suffering, etc. This does not include non-physical forms of suffering, such as suffering from unrequited love, or a lost friendship after treating a person like a jerk.
Why do I make this distinction? Because one of our premises, as requested by so many theists, is that free will is a highly desirable attribute of the universe such that God wants to make beings that have free will (or the illusion thereof, but we are not entering that debate here). Since God is in charge of the physics of the universe, God is culpable for physical suffering, but not culpable for suffering caused by two friends splitting apart as that comes ostensibly from the friends' own free will.
"But Cat," you might ask, "what about physical suffering people inflict on each other, like a stabbing?" Ostensibly God could stop this sort of suffering without removing free will: people physically incapable of stabbing each other are still relevantly free because they can still wake up and choose what they're wearing today, where they're going, who they're hanging out with, what they're going to do when they get there, etc.
"But Cat, someone that can't stab someone else is less free than someone who can." Technically true, but we have to ask if that's a good sort of freedom to have for the sake of it? We already have multiple actions we aren't free to do, yet still consider ourselves free: I can't teleport to Mars or walk unaided on my ceiling. Am I not free because of these few actions I'm physically prohibited from performing? I don't think that would be a reasonable take to have. Additionally, if we really did want to make the argument that "more freedoms, no matter what kind, are better," then we must reach the absurd conclusion that a universe where we're free to enact Hellish tortures with telekinesis or pyrokinesis or whatever on other people only seen in horror movies in our universe is somehow "more free" and therefore "better." I don't think that's a good line of argument, and I hope you agree. So, let us agree that God can cease physical suffering without removing our status as free beings with agency, and that objections this is "less" free are ill-founded.
So, with all of that background established, is it possible for an omnipotent, omniscient, and omnibenevolent being to create a universe with free agents that doesn't contain physical suffering?
Yes. I think it was Swinburne that first called one of these concepts a Toy World, so I will use the terminology here. Can God make a Toy World? How would we know if that's possible?
The argument is simple: if something can be simulated, then an omnipotent, omniscient being can actualize it; because if something can be simulated, that means that it's logically possible.
It's easy to conceive of a universe in which physical suffering is physically impossible. Most gamers have done this in the 90's when they first typed in "iddqd" while playing Doom. Nearly any video gamer that's used an invincibility cheat can probably easily imagine a world in which actors in that world don't suffer physically because the physics of the world simply don't allow it. We can imagine that God could easily do this with something like conditional physics (e.g., for a knife blade, if the knife blade meets tomato flesh, then cut. If the knife blade meets human flesh, remove all inertia).
If God is omnipotent, omniscient, and omnibenevolent, then God should have created a world minimizing the amount of suffering to the extent possible to keep free agency according to the premises. Such a world would be a Toy World, one in which physical suffering is impossible, yet in which free agents can still make choices (and so, they can still suffer if they bring it upon themselves entirely without physics, such as by breaking a friendship or unrequited love: not even an omnimax being could prevent that if free agency is a goal, so God is not culpable).
Yet the world we observe has plentiful sources of physical suffering. This is incongruent with the existence of an omnipotent, omniscient, omnibenevolent God existing: due to God's omnipotence, He could have made the Toy World, due to His omniscience he knew how to make the Toy World, and due to his omnibenevolence he ought to have created the Toy World, but He ostensibly chose not to, and created a world with copious amounts of physical suffering instead. This incongruence is a good reason to doubt the existence of an omnipotent, omniscient, and omnibenevolent God exists.
Now, I mentioned that the definition of omnibenevolence (since it is rather inexhaustive) allows for some theodicies: we can still get our typical theodicies such as the good-making theodicy (maybe physical suffering is a necessary means to a good end, e.g. firefighters are heroes, heroes are good, and firefighters/heroes can't exist unless the bad things they fight exist). My response to this sort of theodicy is that it's kind of like arguing it's "worth it" to invent smallpox because it would be "good" to cure smallpox. It's hard to articulate why this is absurd, but there it is, I hope you agree that this is absurd.
We might also encounter the theodicy by which the theist says, "God actualized physical suffering for a mysterious reason that only God knows, but we can be sure that it's a good reason." This is special pleading, and I always like to illustrate how this sort of thinking is a trap that can never be escaped from. Let me tell you a quick story and I think you'll agree that once this special pleading is adopted, nothing can ever shake the theist out of it, and I think we should agree that these kinds of traps are fallacious for a reason and should be avoided:
For obvious reasons, I think we can see why this kind of special pleading is fallacious and should be warned against.The Special Pleading Problem in Story Form (adapted form a story by Mark Vuletic) wrote: Suppose that P dies and goes to Heaven, and he's thrilled to enter a room with God sitting on a throne. God gets up, cackles wickedly, and mercilessly hits P in the stomach with a flanged mace.
P is hurt, but he thinks to himself, "I know that God is good, and since I am a finite creature, I can't always imagine the reasons God has for doing some of the things that God does. Therefore I can conclude that God had a good reason for doing that, and isn't malevolent."
God strikes P with the mace again. Then again. Then again. Over, and over, and over. P still thinks, "Surely God has some good, yet unknowable reason for doing this to me."
A year goes by. A decade. A century. P still thinks God has a good reason. A millennia. An eon. And so on.
At what point can P ever give up and just admit that God might be malevolent? His thinking -- assuming that his finite epistemology and God's presumed goodness -- is completely unassailable. So on he goes being mercilessly beaten by a malevolent God, forever believing God is benevolent for reasons mysterious to him.
I'll conclude the post at this point. There are a multitude of other theodicies and defense, but I can respond to those as they come out. Thanks!