Which is grammatically correct: "woke up by the..." or "woke up to the..."?

The first one ("by") suggests the cause of her awakening.

The second ("to") suggests that there was an incidental sound at the time she awoke.

They are both correct in common use but have different meanings.

• by: preposition meaning "through the agency or instrumentality of" - he was killed by his attacker
• to: preposition meaning "accompanied by" - they danced to the music

Actually they both seem grammatically correct. 1st one seems to say that she actually woke up "due to the" rattling sound and the 2nd one seems to suggest that she woke up by herself but heard the rattling sound going on.

Yes, both are grammatically correct; it really depends on what you want to express. If you want to say that she woke up because of the rattling, then use the first one. If you want to say that she woke up on her own accord, then use the second one. However, having said that, the first one would be better stated if you said, she woke up "from" the rattling sound.

Except in an extremely contrived context, I don't think OP's first example is valid. Google Books returns just 205 instances of woke by the sound, and it seems to me almost all of those have the verb form wrong. They should be woken by the sound, which itself returns 15,300 hits.

You're woken by something if that thing is what causes you to wake.

You wake to something if that's what you first perceive (usually, hear) when you wake up.

Note that although the by form is normally used in the context of unplanned awakenings, it can be used of a deliberate method - I made sure I woke [up in time] by using a loud alarm clock.

Sophia woke up by the rattling sound of her washing machine.

I don't know what this means. It does not sound right to me. And it certainly isn't idiomatic. I would guess it means one of the following two statements, but I wouldn't know which.

Sophia woke up to the rattling sound of her washing machine.

The sound of her washing machine was the first thing Sophia was aware of when she woke up. (I, myself, would be inclined to omit the up in this example sentence, but perhaps that's just me.)

Sophia was awakened by the rattling sound of her washing machine.

(Bill Franke's suggestion.)
The sound of her washing machine woke Sophia up.

Bonus:

Sophia woke up by her washing machine.

When Sophia woke up, she was beside her washing machine.