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Grandma Mario is a late grandmother of Mario and Luigi's, her own name being Mario's namesake. It is unknown as to whether she is Mario and Luigi's maternal or paternal grandmother. Indeed, Grandma Mia, another grandmother mentioned by Mario, may or may not be the same character as Grandma Mario. Grandma Mario's only role to date is in The Super Mario Bros. Super Show! live-action segment "The Great Hereafter".
The Super Mario Bros. Super Show!
Little is actually revealed about Grandma Mario's past life, except that she was very strict towards Mario and Luigi when they were younger, and was not much of a looker. At some point, she attempted to scold Mario in his daydreams at Mario Brothers Plumbing from beyond her grave in Newark on several occasions. However, she could never fully finish her scoldings, as Luigi would always snap Mario out of his daydreams.
Mario and Luigi soon enlisted the help of The Old Psychic Lady with the Evil Eye Who Reads Fortunes and Knows Everything Before it Happens to help them reach out and call forth Grandma Mario. Eventually, they succeeded, only for Grandma Mario to use The Old Psychic Lady's body as a host. Back from the dead, but in another body, Grandma Mario made her return by smacking Mario and scolding him.
Looking around Mario Brothers Plumbing, Grandma Mario was appalled to see the messy condition the place was in and was unable to describe it in her own words. Luigi suggested that it was a dump, but she instead told him to shut up and not talk back to her. Grandma Mario then exclaimed that their dumpy heart broke her heart and made her the laughing stock of wherever she had been buried, despite the fact that was unable to remember the location, which Mario and Luigi guessed was either Newark or upper Newark.
Grandma Mario then told Mario and Luigi to clean up their apartment, told Luigi to wash behind his ears, and told Mario to trim his nose hairs, so that she could finally rest easy, even if it had to be in Newark. With that said, Grandma Mario fell back onto a chair as her spirit left The Old Psychic Lady's body, presumably returning to her grave.
Other Possible Mentions
There have been other mentions of an unnamed grandmother to Mario and Luigi, meaning that the character in question could be either Grandma Mario or Grandma Mia, or that all these references refer to a single "Grandma" character.
Nintendo Comics System
In the Club Nintendo comic "Super Mario Klemp-Won-Do: Muskeln sind nicht alles!", a grandmother of Mario's is mentioned by the narrator. Mario fought against Blanka, and defeated him by throwing Bananas into the ring, making Blanka slip and fall. It is said that even Mario's grandmother used to advise against keeping banana peels lying on the ground, and that it's "good that Mario never listened to her."
Being a grandmother to Mario and Luigi, Grandma Mario is the mother and mother-in-law of Mama Mario and Papa, although it is unrevealed as to which side of the family she hails from. If she is their paternal grandmother, then her husband would've supposedly been Grandpapa. Mario and Luigi's Aunt Luigeena and Aunt Maria are either her daughters or nieces. Marianne and Luigeena are either her granddaughters or great nieces. Mario Joe and Luigi Bob are either her grandsons or great nephews.