Looking for the name of a fallacy that's similar to the naturalistic fallacy but not quite...

I don't see this as a fallacy, but more of a philosophical concept.

In its basic form: We need [problem], otherwise, we wouldn't have [solution].

In the evaluation of the equation, this does not make sense if the following is true:

1) We agree that the best possible world is the desirable outcome.
2) The non-existence of the problem and solution is better than (brings us closer to the best possible world) the existence of the problem along with the solution.

The reason I don't see this as a fallacy is because perhaps reasonable arguments could be made for the existence of the some problems along with their solutions leading to the better world. So as a general rule, it might be true sometimes, but each case needs to be evaluated. I can't think of any such arguments, so the other option is that the statement is never true. In this case, we simply have a false statement.

As Dr Bo points out, this is a bit of a chicken-or-egg problem. But in some cases, the broken window fallacy does apply, for example, when opportunity costs are overlooked.

Alice: Why the hell did you throw a brick through Mrs Pratchet's window?

Bob: Because now she has to pay someone to replace the window, so the window-maker will earn money, and if they spend that money, that goes back into the local economy!

Alice: ...you're a moron. And here come the cops.

Bob is overlooking the fact that:

1) Mrs Pratchet has to spend money replacing something that was broken, rather than buying something new

2) The window was just fine beforehand - so he destroyed a perfectly good item for nothing

3) He's caused Mrs Pratchet undue distress

4) Throwing bricks through people's windows is pretty illegal.

In some cases, you might be able to make a consequentialist argument - assuming the net benefit of doing X far exceeds the costs, you could claim the ends justify the means. In terms of problem-solving, this might be the case if the resources developed to tackle a problem can be used to deal with other problems (e.g. technology used in one industry can be deployed in another). This isn't an impregnable claim - you could easily say that a) the ends never justify the means, or b) the ends might justify the means, but in this case, they don't.

Since the argument talks about necessity, this would assume that they know for sure bad things cause good things, and have some kind of overall calculus on the good-to-bad ratio. Why do I say it like this, all complicated? Because ethical claims carry moral principles with them. So I think our first task would be to identify what principle they're using. If what I think they're saying what they're saying is correct, then they would be committed to consequentialism. 

One major critique of consequentialism is precisely that it demands the forekowledge of consequences. And one big assumption in normative ethics is that a plausible ethical theory cannot demand the impossible. 

One fallacy that I can think of is the historian’s fallacy. This is because the claim made on necessity depends on what they already know: certain diseases led to medical advancements. It does, however, sound plausible that new diseases may led to new discoveries, but are they necessary for those discoveries? I honestly don't know. If so, then there's no fallacy and I'll retract my charge.

I think the part that follows answers your question more directly

In a more general sense, we could provide an example were bad things need not cause good things. 

Example: A person dies and their spouse never recovers from sadness. 

If we're counter with "you don't know what other good things could had come and increased the good-to-bad ratio" then this would be an argument from ignorance.  

For your statement "It's basically thinking that a problem needs to remain in order for us to still have the solution when the solution has absolutely no purpose outside of its relation to the problem."

I guess the principle would be that the origin of the good thing is necessary for its continual existence. This is factually incorrect.

Example: The couple started quarreling and that brought them together. Therefore, they should keep quarreling to stay together.

The evidence is irrelevant. The fact that the conflict brought them together bears no evidence in the claim that they should stay at it. This is a non sequitur in the informal sense.

Any fallacies where there are no hidden assumptions?

Reading your example again, it seems that the arguer is using what is known to be a tautology. We can unveil it by phrasing it like this: sickness is necessary to have cures for them. Therefore, sickness is good for having cures. And by 'good' we mean that medicine is good because they cure. This is circular reasoning. The proposition is that sickness is a good thing because they develop medicine. And the development of medicine is good because they cure.