Mystery fanfiction is one of the few genres that carries a lot of weight. Different from fluff mystery needs a plot that will keep the readers on the edge of their seats. It has to be memorable and make people want to comment their thoughts.

For a mystery fanfiction it is most likely impossible to write without a prompt. It is because a mystery should bring question to your mind. If you do not have a prompt and just randomly start writing you might run out of questions and run out of ideas to continue the story. 
If you have a prompt and thus a good plotline, ideas will keep overflowing. (This is different for everyone it can be that you are one of the people that don't need a prompt.)

How do you find a prompt? This can be found by 
1. Listening Music. 
2. Looking at the translation of lyrics. 
3. Quotes.
4. Looking at the place around you.
5. Looking at videos
6. Looking at pictures.
7. And a lot of more things.

One way to come up with a plot is like this. I will use the lyrics translation as example.
Let me make sure that you know that I mean just giving a VERY SIMPLE example. You should try to be more creative with this. 

1. Sometimes you are just curious about the translation of a kpop song. Most of us aren't Korean so they might look it up. For example I looked up shinee's hello and saw this:

"This is not the first time
Truthfully I've loved and parted
But i'ts hard, please believe in me
You are different"

2. This might bring a lot of questions to your mind like:

How am I different?

With whom have you parted?

Why have you parted when you loved? (Remember that love doesn't mean romance love can mean like love your mother)

How many times have you loved and parted?

3. See a lot of questions already. Now start trying to answer your own questions using your fantasy.

I'm not human and he is. or I'm human and he isn't 

He has parted with all his lovers. or He has parted with all the people he got close to. 

I'm dangerous I can hurt everyone around me I have to be careful. or People don't like me because I don't look normal, I don't understand why. 

4. Put all of the questions together and write the answers down somewhere where only you can find it. 
Form the plot and start answering the questions bit by bit, don't answer them all at one. 

This is one way on comming up with a plot, it is not the best. I will do some research on better ways. You will know when I have found more ways to come up with a plot.