Sometimes, the character will wake up and find out that there's something odd with his right hand or the entire arm, or that a mark just appeared over it's palm.
Either way, the character's right hand (or left) has a mysterious and strange power, it can do things that makes him unique from the rest of the cast, or it has a some unknown important purpose.
Some characters developed it's own version via technology. There can be some cases that the power is granted by God or the Evil.
See also Evil Hand
when the character gets a transplanted Evil hand (or limb)
Examples in Anime and Manga
To Aru Majutsu no Index
Kamijou Touma whose right hand, the Imagine Breaker, has the ability to negate any supernatural powers, including the effects of psychic, magic, and divine powers.
Yumeji Fujiwara, also has the ability in daydreams to create weapons or replicate abilities he has seen with his right hand.
Shu Ouma the main character, whose right hand holds a special power called the Power of Kings.
Examples in Video Games
Megaman Zero 4
Zero has the "Z Knuckle", a weapon chip implanted in Zero's hand that allows him to steal the enemy's weapon and use it as his own.
Gene's right hand is one of the legendary God Hands he receives from Olivia.
Examples in Comic Books
The titular Claw The Unconquered
Examples in Film:
, in District 9, undergoes a startling transformation and gains the ability to manipulate alien technology