У героини появилась догадка, но доказательств ещё не было.

Because Russian often uses у + genitive to express that someone has something or that something appears to/for someone.

So у героини появилась догадка literally feels like:

• at the heroine
• there appeared a guess/idea

Natural English: The heroine had a suspicion / a thought occurred to the heroine.

This pattern is very common in Russian:

• У меня есть книга = I have a book
• У него появилась надежда = He got hope / hope appeared in him

So у героини does not mean physical location here. It marks the person connected with the new thought.

Героини is genitive singular of героиня.

It is genitive because the preposition у requires the genitive case when it means at/by someone in this possessive-existential structure.

• nominative: героиня
• genitive: героини

So:

• у героини = with the heroine / for the heroine / the heroine had

Героиня can mean:

1. heroine in the literary sense
2. the female main character of a story

In this sentence, it most likely means the heroine / the female character.

A native English speaker might think of heroine only as a brave female hero, but in Russian героиня is also very commonly used for a female protagonist.

Появилась is the past tense of появиться, which means to appear, to arise, to come into being.

Here it means that a thought or suspicion came to her or arose in her mind.

So:

• У героини появилась догадка = The heroine got a suspicion / an idea occurred to her

This sounds more dynamic than simply saying she already had one.

Because the grammatical subject is догадка, and догадка is feminine singular.

In the clause:

• догадка = subject
• появилась = verb agreeing with that subject

Russian past tense agrees in gender and number:

• masculine: появился
• feminine: появилась
• neuter: появилось
• plural: появились

Since догадка is feminine, the verb is появилась.

Догадка is a tricky word because it can overlap with several English words:

• guess
• hunch
• suspicion
• inkling
• idea one arrives at

It usually suggests that someone has figured out something partly or intuitively, even if they cannot prove it yet.

That fits well with the second half of the sentence:

• но доказательств ещё не было = but there was still no evidence

So the meaning is something like:

• she had a suspicion
• she had an idea about the truth
• but she could not prove it yet

Yes, but the meaning would be a little different.

• У героини была догадка = The heroine had a suspicion
• У героини появилась догадка = A suspicion came to the heroine / The heroine suddenly got a suspicion

So появилась emphasizes the moment of emergence.
Была just describes the state.

If the author wants to show that the thought arose at that moment, появилась is better.

Because after negation in an existential sentence, Russian very often uses the genitive.

Here:

• доказательство = evidence, proof
• genitive plural = доказательств

So:

• доказательств ещё не было = there was still no evidence

This is a standard Russian pattern for saying that something did not exist / was not present.

Compare:

• Были доказательства = There was evidence
• Доказательств не было = There was no evidence

The genitive after negation is especially common in this kind of there was / there wasn’t statement.

Because this is an impersonal/existential construction with быть.

In Russian, expressions like:

• не было времени
• не было денег
• не было доказательств

very often use было in the neuter singular, even when the noun is plural in meaning.

The point is not so much the evidence were not there, but rather there was no evidence as a fact of existence.

So доказательств ещё не было is the normal idiomatic form.

Here ещё means yet or still.

So:

• доказательств ещё не было = there was still no evidence / there was no evidence yet

It suggests a contrast:

• a suspicion had already appeared
• but evidence had not appeared yet

This word is very common for that idea of not yet or still not.

Но means but.

It sets up a contrast between:

• У героини появилась догадка = she had a suspicion
• доказательств ещё не было = but there was no evidence yet

So the sentence means that she may have been on the right track, but she could not support her idea with proof.

The word order is natural and expressive, but Russian allows some flexibility.

This version:

• У героини появилась догадка, но доказательств ещё не было

sounds smooth and literary. It introduces:

1. the person involved (у героини)
2. the new thought (появилась догадка)
3. the contrast (но)
4. the missing proof (доказательств ещё не было)

You could rearrange parts for emphasis, but the given order is very natural.

For example:

• Догадка у героини появилась, но доказательств ещё не было
  This puts extra emphasis on догадка.

But the original is the best neutral/literary version.

In dictionaries, доказательство often corresponds to proof or piece of evidence.

In the plural or in a genitive-plural form like доказательств, it often matches English evidence very naturally:

• нет доказательств = there is no evidence
• нашли доказательства = they found evidence

So although Russian uses a countable noun form, English often prefers the uncountable word evidence.

That is why доказательств ещё не было is best translated as there was still no evidence, not necessarily there were no proofs.