It could be an enemy from one of his past cases. Someone who holds a grudge against L for getting them caught or ruining their plans before.
In some fanfictions, it could be a mysterious organization. They might have heard about L's incredible intellect and want to use him for their own evil purposes. They could be involved in illegal activities like blackmail or espionage and think L can be a valuable asset, so they kidnap him to force him to work for them.
It could be a rival guild member who wants to use Juvia as leverage against Fairy Tail. Maybe someone jealous of her powers or her relationship with Gray.
In some 'damian gets kidnapped fanfiction', the kidnapper could be an otherworldly being. Let's say an alien or a magical creature that has an interest in Damian for some strange reason. It could be that Damian has a unique energy or power that the otherworldly being wants to study or harness. Or perhaps it's a human who has been corrupted by some dark magic or technology and is acting on behalf of a more powerful, hidden force that wants Damian for its own nefarious purposes.
Maybe it's a group of poachers who are after rare Pokémon. Gengar, being a relatively unique and powerful ghost - type, would be a valuable catch for them. They could plan to sell Gengar on the black market for a high price.
It could be Shredder. He's always been after the turtles.
Another option is that it's a human who has somehow discovered the existence of Amethyst and the other Gems. This human could be a mad scientist type who wants to study Amethyst's unique gemstone and abilities. He or she might not understand the full implications of what they're doing and just see Amethyst as an interesting specimen to experiment on.
Perhaps some remnants of Homeworld's military that still hold a grudge against the Crystal Gems. They might think that by kidnapping Amethyst, they can force the Gems to stop their activities on Earth or give up some valuable information.
It could be a rival spy agency. They might see Moneypenny as a valuable target to get information about MI6's operations. They would likely be highly trained and well - equipped.
Maybe it's a character from one of the games Matpat has theorized about. For example, if he made a theory about a game villain being more complex than thought, that villain could come to life in the fanfiction and kidnap him to prove his own power and intelligence.
The kidnapper might be a deranged Toad. Maybe this Toad has been corrupted by some evil force and decides to kidnap Luigi for reasons only known to the twisted mind of this new 'evil Toad'.
It could be Dr. Eggman. He's always after Sonic and would love to get his hands on him to stop Sonic from foiling his evil plans. He might want to study Sonic's speed and use it for his own machines.