In order to answer this question, you should start by identifying the type of story you want to create. Is the story character-driven or plot-driven? Character-driven fanfiction would favor unique characters impacting the original plot: their actions directly changing the course of the story. Plot-driven fanfiction would favor original characters being thrown into a unique story (with or without a proper explanation of why or how, though some fans may feel repelled by the handwave at these questions.) If you already have a full cast of unique characters and settings, you may not be writing fanfiction at all, but a unique work of fiction drawing inspiration from other material.
The truth is, there is no such thing as a unique story or unique character trait, or combination of traits. Everything you can imagine has already been thought up, and probably already produced. What we are left with is the ability to throw unoriginal characters into unoriginal stories. The task of a artist is to find a way to make it our own.
How to do this is impossible to give a matter-of-fact answer to. It is something each and every artist has contemplated (oft in frustration) and all have failed to produce a definitive answer. The best advice I can give is this: Decide whether you want to write a character-driven or plot-driven fanfiction, then work backwards.
For character driven stories, ask yourself what type of character would be needed to alter the original story in the way you want it to go. Build your character(s) around that goal alone, then add in traits later that contradict the social convention, or "cliche," of that character type. (Is he/she a super baddie that forces the original baddies to team up with the heroes? Maybe he could also have a soft spot for cats, or a tendency to do volunteer work - out of a genuine desire to help out.) Making a round character in any work of fiction is possible by doing this, and is recommended by many well known authors.
For plot-driven stories, ask yourself what type of situation the original cast could be in (believable or otherwise) that would produce an entertaining result. This is the most common method in fanfiction I have read, and is arguably the most entertaining when done right. This type of fanfiction does not require any new characters. If it does (say they are transported to a different world, and the new world is inhabited), the characters will very likely not be necessary to develop beyond generic characters, often called "stock" or flat characters. The point of plot-driven fanfiction is to create a unique scenario for the fandom to see their favorite characters in, not to fall in love with new characters (the latter is best used for character-driven stories).
In either case, the most important (and compelling) aspect of fanfiction is to portray the source material in a new light. If the plot is different, what unique factors or characters were responsible for this variation? If the original characters are changed in some way, what unique conditions made them change in this way? The most important aspect of writing quality fanfiction is leaving no deviation unanswered (short of handwave dismissals of how they ended up knowing each other in highschool, despite canon dictating they met later in life, for example. In this case, the fact they met in highschool should be pivotal to the story in some way. An example of this being the fanfiction takes place in highschool, and the character needs to be there in order to tell the story you want to tell. Readers will likely understand the necessity to handwave this type of deviation, and are less likely to be put off by it.)
If you cannot naturally produce scenarios or characters of your own, try playing around with other existing universes (including our own.) Drop the source character(s) into any other universe. Throw characters from another universe into the source material's universe. Does any of these exercises spark ideas you would like to play with? Take one (or more) of them then run with it.
More than likely, you will end up coming up with some ideas of your own along the way, simply by trying out new things. Remember, at the end of the day, nothing is every truly unique, and that's okay. The goal of writing is to tell a story that has been told a million times before, nuanced by the unique perspective of the writer.
If none of the above methods gets your proverbial gears turning, you might need to ask yourself the most important question of all: "Why am I writing this story?"
What are you (or readers) intended to gain by the time lost writing (or reading) your story? Be it intended only for the writer, or for the audience as well, writing is a poor use of your time if it serves no other function than "for the sake of writing." The same is true for writing with the sole intent of making money, and for the same reason: If you do not feel passionate about what you are writing, you will have nothing unique to bring to the table.
Contrastly, when you write about things you know and have an opinion on, you have access to an endless archive of ideas to draw from, because it's the same things running through your mind every day. Answer that fundamental question, and make sure the purpose is being pursued in your writing. Do that, and you should have no trouble in coming up with ideas for your fanfiction.
Final thoughts: If you do not care about public approval, and;
A) write because you want to hone your writing ability, you don't need to come up with a unique story for that. It can be done using even the most cliche of plotlines and characters.
B) write simply because you enjoy coming up with cool ideas, you do not need to ever write a single word for that. The "fun part" of writing isn't the writing itself. It's the brainstorming. You can do that all day long (as I often do) without ever picking up a pen!
- Decide whether you want to write a character-driven or plot-driven fanfiction.
- Change only what is necessary to your story (fanfiction), or keep only what is necessary to your story (fiction).
- Draw inspiration from anything and everything.
- Know why you want to write what you want to write.
- Decide whether writing is a viable use of your time.