The kidnapper could be an estranged family member of Beckett. There might be some family feud or inheritance issue that leads them to believe that by kidnapping the baby, they can gain an upper hand. Or it could be a completely random person who has a mental disorder and randomly targeted Beckett and the baby because they saw them as an easy target.
The parents of the Beckett baby would surely be main characters. They would be in a state of panic and desperation, driving the story forward with their search for their child.
One possible plot for a 'beckett baby kidnapped fanfic' is that the kidnapper leaves behind strange clues. These clues lead the family and the authorities on a wild goose chase at first. As they start to piece things together, they realize that the kidnapping is part of a larger conspiracy. The family has to face many challenges and overcome their own personal demons while trying to get the baby back.
One possible kidnapper could be a rival engineer who is jealous of MacGyver's fame and ability to solve problems so effortlessly. This person might think that by kidnapping MacGyver, he can steal his ideas and take credit for his inventions. Also, it could be a group of eco - terrorists who believe MacGyver's work has been harmful to the environment in some way and want to force him to make amends by using his skills for their cause. A third option is a deranged fan who has an unhealthy obsession with MacGyver and wants to keep him all to himself, thinking that kidnapping is the only way to have MacGyver's undivided attention.
Perhaps a dark guild that has been observing Fairy Tail for a long time. They see Mirajane as a valuable asset and believe that her powers can help them in their evil plans. So they plot to kidnap her and force her to cooperate.
It could be a group of rogue Death Eaters who want to use baby Harry as a bargaining chip against Voldemort. They might think that by having Harry, they can gain power within the dark wizard's ranks or even overthrow him.
It could be a jealous rival from his past. Maybe someone he outperformed in a competition or a former acquaintance who has a grudge against him.
It could be a group of rogue shinobi from a neighboring village who are jealous of Konoha's power and see Hinata as a valuable hostage. They might think that by kidnapping her, they can gain some leverage over Konoha in terms of resources or territory.
A powerful and vengeful warlord from a forgotten realm could be a kidnapper. He might see Wonder Woman as a threat to his plans of conquest and decides to take her out of the picture by kidnapping her.
It could be Orochimaru. He has always been interested in Naruto's power and potential. He might kidnap Naruto to try and extract the Nine - Tails' chakra or study his unique Uzumaki bloodline further. Another possibility is a group of rogue ninjas who want to sell Naruto to the highest bidder, thinking his value as a Jinchuriki would make them rich.
One possible motive could be power. If Naruto has some unique abilities or is a key to something powerful in the fanfic world, the kidnappers might want to control him to gain that power for themselves.