"Good" can be a difficult term to nail down exhaustively. A better way to look at it would be to put it in terms of suffering: rather than asking if God is "good," we could instead simply ask if God is malevolent -- that is, does God ever knowingly and intentionally share culpability for gratuitous suffering by a) potentiating it, b) causing it through action, or c) allowing it through negligence?
Many monotheists have the intution (or some say support from holy texts, or whatever) that the deity is at least not malevolent -- this is usually entailed by being "perfectly good," but we can avoid hairy definitions by just focusing on God being "never-malevolent."
Given the deity is omnipotent (has the capacity to actualize any logically possible state of affairs by fiat), omniscient (at least knows all possible truths and believes no falsities), created the cosmos and all of its apparent contingencies, and that the deity is never malevolent; then there is a contradiction with the existence of suffering in the actual world due to God being culpable for it in terms of (a), (b), or (c) listed above.
Here's why, if we assume the deity has the properties listed above:
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a) Culpability through potentiating
What I mean by "potentiating" is this: the actual, existing world -- this one, according to the premises the one that God deliberately created -- has what appear to be contingencies that could have been otherwise. For instance, while there could never logically be a Euclidean square-circle, there is no apparent contradiction with a possible world where gravity repels instead of attracts: God could have made such an anti-gravity cosmos, but instead ostensibly to create a universe where gravity attracts. Just as a silly example.
The way this relates to culpability for suffering is because if God is omniscient, then God knows all different iterations of all possible worlds (all different arrangements of logically possible contingencies), and in order to actually create one of them (as, according to the premises, God did -- this one), it must be the case that God is culpable for the ramifications of its choice of contingencies since, being omniscient, God knowingly chose that specific set to make real rather than any of the other possible ones.
A simple analogy is a carpenter who could build a house for someone who depends on him -- he has a choice among an infinite number of possible blueprints. Some are mansions, some are shacks, some don't have doors or windows, others are weirder still. Now let's say the family is totally dependent on the carpenter for their living space for some reason -- they have no other choice other than to live in whatever the carpenter builds, and the carpenter up and decides to build a house without a roof.
Is the carpenter culpable for the family getting sick or wet and miserable when it rains, given that the carpenter knows they have no other option than to live in the home he builds, and given that the carpenter knows that by leaving out a roof he has condemned them to unnecessary pain? Yes. The carpenter has potentiated their suffering by causing to exist a capacity for it in the very nature of their surroundings as a creative being, where there are otherwise other options which avoid said suffering (such as choosing blueprints with a roof).
So, back to God being the omnipotent, omniscient, never-malevolent creator of the cosmos. Consider something like... oh I don't know, hurricanes, or tornados, or simple discomfort from the cold, or leukemia. It's logically possible for there to be a world where these things don't happen, yet in which people could otherwise live out their lives just fine. This is essentially God neglecting to build a roof over the house it built for us, and so God is culpable for potentiating suffering.
(This is also the case for violence and most other "free will" attributable sources of suffering -- God still shares some culpability in that. However, that's a more convoluted argument, and for now it's sufficient to just understand what it means for God to be culpable through potentiating.)
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b) Culpability through action
Obviously a being is at least sometimes malevolent (and thus not "never-malevolent") if said being directly causes or sets in motion sources of suffering knowing full well that they would entail such. I don't see (b) being very controversial for that reason: if the theist simply shrugs and says "yeah, God is vengeful and wrathful etc." then they're simply implicitly dropping the original presupposition that God isn't malevolent.
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c) Culpability through negligence
This is tied pretty closely to (a), since in order for God to negligently allow suffering, it must first be potentiated in the first place. Thus any negligence attributable to the deity is something of a "double whammy" of culpability for the suffering in question.
Another silly analogy time: consider that the universe hypothetically consists of two rooms filled with people going about their business. Assume these two rooms are paradises such that there are no sources of suffering created in either of them. Now let's say I drop by one of them and place a loaded pistol in the middle of the room somehow, safety off.
First, note that I have just potentiated suffering already where previously there was none, and where otherwise none needed to exist. In one sense, I'm already guilty -- already culpable at least in part for anything that happens.
Secondly, since I know full well what could happen if someone were to pick it up and have an accident or get angry in an argument or something -- yet I left it there anyway -- I'm culpable in part for the suffering again through my negligence.
Now consider the contrast between the two rooms: one has a loaded pistol, the other does not. One has potentiated suffering (even if no one has picked up the gun yet, now there is the potential for it due to my actions) whereas the other does not.
Why on earth would any sane, rational, non-malevolent being up and decide to potentiate suffering when it otherwise doesn't have to be potentiated?
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Thus, given the premises, it's inescapable without engaging in fallacy that there is a contradiction with the premises that God is omnipotent, omniscient, creator of the cosmos, never-malevolent, AND that suffering of various types are for some reason potentiated in the actual world by a being which ostensibly otherwise could have chosen not to potentiate them.