Well, it could be that he's been injured in a battle against an akuma. Maybe the akuma had a particularly powerful attack that left him unconscious and motionless.
Perhaps he's under the influence of some sort of magic or spell in the fanfiction world. It might be a side - effect of an experiment gone wrong by a villain who was trying to use his powers for their own gain, leaving Adrien in this state.
Ladybug might come to his rescue. She could use her Lucky Charm to find a way to heal him or break whatever spell he's under.
It could also be due to a spell or an akuma's special ability. In the Miraculous universe, there are many powers that can immobilize characters. For example, if an akuma had the power to paralyze people, Adrien might have been caught in it. And since he's often in the middle of the action, he'd be a prime target. He might be motionless as his body is fighting off the effects of this external force, trying to break free from it so he can transform into Chat Noir and save the day.
It can create a sense of suspense. The other characters will be worried about him and the readers will be eager to know what will happen next. For example, if Ladybug sees him motionless, she'll try to figure out what's wrong and how to help him, which drives the plot forward as she investigates.
One possible reason could be that he got injured during a fight with an akumatized villain. In the Miraculous Ladybug universe, battles can be quite intense and Adrien, as Chat Noir, often puts himself in harm's way to protect Paris. Another reason might be some sort of illness that was magically induced, perhaps by a new villain with a different kind of power that targets the heroes in a non - traditional way. It could also be due to overexertion from using his Cat Miraculous too much without proper rest, which could lead to his body breaking down and him ending up in the hospital.
One reason could be that he's in a state of shock. Maybe he witnessed something really traumatic that made him unresponsive. For example, if he saw Ladybug in a life - threatening situation and couldn't do anything to help immediately, it might have sent him into a state of mental paralysis.
In an Adrien's mom fanfiction, she might have left behind secrets. For example, she could have hidden some important Miraculous or knowledge. The story could revolve around Adrien discovering these secrets and learning more about his mother, which in turn helps him in his battles as Chat Noir.
One possible plot could be that Marinette, in an overzealous attempt to get closer to Adrien, creates a 'trap' of sorts. Maybe she arranges for a photoshoot at a location where only she and Adrien are present, under the guise of it being a school project related to fashion. She could have set up the set in a way that Adrien would be impressed by her creativity and they would have an extended period of time to talk and bond.
Well, the plot could involve Adrien initially being confused by the power of the Butterfly Miraculous. Maybe he accidentally activates it and starts sending out akumas without really understanding what he's doing. Then, he has to figure out how to reverse the chaos he's caused while also hiding his new - found power from his friends and family.
One possible plot could be that Adrien stumbles upon the Peacock Miraculous by accident while exploring an old attic in his mansion. He then has to figure out what it is and how to keep it safe from Hawk Moth.
The plot could involve Adrien being akumatized after a big fight with his father. He might feel extremely alone and angry. His akumatized form could have the ability to freeze time in certain areas. Ladybug would struggle to figure out how to defeat him without hurting him too much. She might have to team up with some of the other Miraculous holders like Rena Rouge and Carapace to come up with a plan.