A lot of Pokémon theories seem to revolve around potentially dead characters. A popular one is the "Ash is dead" theory, but there are also a lot of theories about Ghost-type Pokémon being dead versions of preexisting Pokémon. This line of thinking begs the question of what Pokémon like Mimikyu might have been.
One prevailing theory suggests that Mimikyu is actually a dead Pichu. This sounds improbable at first, but there's logical reasoning behind it. Looking at all the available information for Mimikyu makes it possible to conclude that the Disguise Pokémon may have been a neglected Pichu in life.
To understand the theory better, some key information regarding Mimikyu is necessary. Supposedly, anyone who catches a glimpse of Mimikyu under its disguise will either die of shock or be attacked mercilessly by Mimikyu. Whatever's under the Pikachu disguise would seem to prefer that others see it as a Pikachu.
Mimikyu's desire to be like Pikachu comes from loneliness. It saw that Pikachu merchandise was extremely popular, so it redesigned the rag covering its body to more closely resemble the electric mouse. Unfortunately, its efforts only made it look even creepier and drove more people away. This is the tragedy of Mimikyu-- a Pokémon that only wants to be loved but can never get it.
Mimikyu's lack of attention causes it to hold Pikachu in utter contempt. As the anime shows, any Pikachu unfortunate enough to meet Mimikyu's gaze will be relentlessly attacked by the Disguise Pokémon. These attacks are likely the result of the burning jealousy felt by Mimikyu.
Knowing everything about Mimikyu's disguise and why it hates Pikachu makes it possible to theorize on how it can be a dead Pichu. Looking at it in a meta sense, Pichu can be said to be a less-popular version of Pikachu among the Pokémon fandom. It shares this quality with the other electric rodents made to resemble Pikachu in each generation such as Pachirisu, Emolga and Togedemaru. There's even a variation on this theory that suggests that Mimikyu is actually a deceased Dedenne. All of these Pokémon have the right to be mad at Pikachu for stealing their thunder.
Another way Mimikyu could be a Pichu is in its semblances to the Pikachu family. Besides its outward appearance, Mimikyu learns several moves that the Pikachu line can learn; Thunderbolt, Thunder and a couple more Electric-type moves are of particular note. Its in-game cry also sounds eerily similar to Pikachu's anime cry. These could all be taken as either superficial similarities or hints to an unstated connection.
There's also something to be said for Mimikyu's typing. As with most dead Pokémon theories, it's likely that Mimikyu switched its primary type from Electric to Ghost when it passed on. The Fairy typing seems a little trickier to argue, but the Pikachu line is a part of the Fairy Egg Group; again, Mimikyu may have changed to the Amorphous Egg Group upon becoming a ghost. By this logic, Mimikyu could reasonably have been a Pichu that died and went through some serious-but-not-unreasonable changes.
Pichu's evolution method could also be taken as a hint for why Mimikyu is so malicious. A normal Pichu is able to evolve once they've reached a high enough friendship level. Since Mimikyu is a lonesome Pokémon, it's possible that it was originally a Pichu who never had a significant friendship in life. If Mimikyu has truly never felt proper love before, its willingness to instead bear a grudge makes all the more sense.
If Game Freak or The Pokémon Company were ever to come out and admit that Mimikyu is a dead Pichu, there would be plenty of evidence to support it. Mimikyu's Pokédex entries, personality, move set, typing and similarities to the Pikachu line all suggest that the two are closer than anyone's letting on. Given everything this Pokémon theory has going for it, nobody can be faulted for adopting it as their official headcanon.
