How did Murph figure it out?

In the scene where Murph is an adult and is in the room trying to figure out the puzzle of what was communicating through the books, she gets the 'aha' moment where she realizes that it is her father somehow communicating. The 'stay' message is crucial to her figuring this out. When she first examines the watch she only gives it a quick glance and sees that the second hand is moving but it's not ticking, which commonly happens with clocks when the battery is low but the clock isn't dead.

Once it clicks with Murph that it's her father trying to communicate she takes another look at the watch. On closer inspection, she sees that the 'ticking' is irregular and recognizes it as Morse code. In an earlier scene with young Murph we see her trying to decode the book messages using Morse code, so we know she's familiar with it.

Once she starts interpreting the message using Morse, she immediately recognizes that this is indeed a message, and continues decoding it, giving her the data she needed to solve the gravity equation.

If this was the scene you found too far-fetched, well I don't know how you got through the rest of the movie. The whole thing is far-fetched, and that ambition is what makes it great, in my opinion.

I think tarnold86 gives a great answer. In addition to their answer I would say that Murph's father uses "love" to find Murph in the tesseract at the right times, because love transcends time and space:

Brand: Maybe it means something more - something we can't yet understand. Maybe it's some evidence, some artifact of a higher dimension that we can't consciously perceive. I'm drawn across the universe to someone I haven't seen in a decade, who I know is probably dead. Love is the one thing we're capable of perceiving that transcends dimensions of time and space. Maybe we should trust that, even if we can't understand it. All right Cooper. Yes. The tiniest possibility of seeing Wolf again excites me. That doesn't mean I'm wrong.

Cooper: Honestly, Amelia... it might.

So in the Interstellar universe love is able to cross the divide between Murph and Cooper, and as a result of love crossing the divide Murph "feels" that the ghost is her father.

Young Murph: Dad?

Cooper: Sorry, Murph. Go back to bed.

Young Murph: I thought you were the ghost.

Cooper: There are no such things as ghosts, babe.

Young Murph: Grandpa says you can get ghosts.

Cooper: Maybe that's because Grandpa is a little too close to being one himself. Go back to bed.

This was also foreshadowed and obliquely hinted to Murph:

Cooper: After you kids came along, your mom, she said something to me I never quite understood. She said, "Now, we're just here to be memories for our kids." I think now I understand what she meant. Once you're a parent, you're the ghost of your children's future.

Also Murph always thought the ghost was a person trying to tell her something:

Getty: You have an idea?

Murph: A feeling. I told you about my ghost. My dad thought I called it a ghost, because I was scared of it. But I was never scared of it. I called it a ghost, because it felt... it felt like a person; like it was trying to tell me something. If there's an answer here on earth it's back there, somehow in that room.

As you can see in that quote Murph has "a feeling." I think her intuition that she had as a child that her ghost was her father, was confirmed, in her mind, when she found the watch her father gave her encoded with data. This further would be reinforced by her desire to believe in the nobility of her father which was put into doubt when she suspected her father abandoned her. The watch proved to her that her father really was trying to save her, and humanity. I think the bond of love made her initially suspect that her ghost was her father, it caused the "feeling" she had to reenter the house, and finding the data encoded in the watch confirmed her suspicion.

In the end humanity was saved because of the powerful love between a father and a daughter. That same love is why Murph was able to figure out her ghost was her father.