Her derdin bir çözümü var.

In Turkish, her is a completely different word from English her. It means every or each.

So:

• her dert = every problem
• her gün = every day
• her insan = every person

It is just a coincidence that it looks like the English word her.

This is one of the most useful things to notice in the sentence.

derdin comes from dert (problem, trouble) and has the genitive ending:

• dert → base form
• derdin = of the problem / of the trouble

Why is that needed? Because çözümü means its solution or the solution of it. Turkish often expresses this kind of relationship with a genitive + possessed noun structure:

• derdin çözümü = the problem’s solution / the solution to the problem

So:

• Her derdin bir çözümü var
  literally: Every problem has a solution
  more literally: For every problem, a solution exists

Because of a common sound change in Turkish.

Some words ending in -p, -ç, -t, -k often soften when a vowel-initial ending is added:

• t → d
• k → ğ or g
• etc.

So:

• dert
  • -in → derdin

This is very normal in Turkish. Compare:

• kitap → kitabı
• ağaç → ağacı
• renk → rengi

çözümü is:

• çözüm = solution
• çözüm-ü = its solution / the solution

That -ü is the 3rd person possessive ending. It shows that the solution belongs to something.

So:

• çözüm = solution
• derdin çözümü = the problem’s solution / the solution to the problem

This is why derdin and çözümü go together. Turkish often uses this pattern where English might simply use of or to.

No. In çözümü, the ending is part of the noun.

• çözüm = solution
• çözümü = its solution / the solution

It is not the separate yes/no question particle.

For example:

• Bu çözüm mü? = Is this a solution?
  Here mü is separate.

But in:

• bir çözümü var
  the -ü belongs to çözüm and is not asking a question.

Here bir is best understood as a or one in the sense of some solution.

So:

• bir çözümü var = it has a solution / there is a solution

In many contexts, bir works a lot like an indefinite article, even though Turkish does not have articles exactly like English.

Compare:

• çözüm var = there is a solution / solutions exist
• bir çözüm var = there is a solution / there is one solution
• bir çözümü var = it has a solution

In this sentence, bir helps give the natural meaning a solution.

var means there is / exists.

Turkish often uses var and yok to express existence or possession.

So instead of saying has directly, Turkish often says something closer to:

• X’in Y’si var = X has Y
• literally: X’s Y exists

That is exactly what is happening here:

• Her derdin bir çözümü var
  literally: Every problem’s a solution exists
  natural English: Every problem has a solution

Right. Turkish usually expresses have with var.

Examples:

• Arabam var. = I have a car.
• Vaktim yok. = I don’t have time.
• Onun bir fikri var. = He/She has an idea.

So in your sentence:

• Her derdin bir çözümü var = Every problem has a solution

This is a very common Turkish pattern.

Because Turkish and English build possession differently.

English says:

• Every problem has a solution.

Turkish says something more like:

• Every problem’s a solution exists.

So the Turkish order is:

1. Her derdin = every problem’s
2. bir çözümü = a solution
3. var = exists / there is

This is perfectly normal Turkish word order.

Yes, you can, and it is understandable.

• Her derdin çözümü var = Every problem has a solution.
• Her derdin bir çözümü var = Every problem has a solution.

Adding bir can make the idea sound a bit more like a solution or some solution, which fits the English sentence nicely.

So both are possible, but bir çözümü var is very natural.

In Turkish, her is normally used with a singular noun, even though the meaning is every.

So:

• her gün = every day
• her insan = every person
• her dert = every problem

This is different from some English learners’ expectations, but it is standard Turkish.

Yes, in another context derdin can also mean your problem/trouble:

• dert-in = your problem

But in this sentence, because of her and the whole structure, it is understood as the genitive form:

• her derdin = of every problem

So here it does not mean your problem.

Yes. vardır is also correct.

• var = neutral, common, everyday
• vardır = a bit more formal, assertive, or written

So:

• Her derdin bir çözümü var. = normal everyday statement
• Her derdin bir çözümü vardır. = slightly more formal or emphatic

Both are grammatical.

A useful breakdown is:

• her = every
• derdin = of the problem / of each problem
• bir = a
• çözümü = its solution
• var = exists / there is

So the sentence is built like this:

• Her derdin bir çözümü var.
• literally: Every problem’s a solution exists.
• natural English: Every problem has a solution.

This is a great example of how Turkish uses genitive + possessive + var instead of the English verb have.