From the Super Mario Wiki, the Mario encyclopedia
Grandma Mario is a late grandmother of Mario and Luigi. Given their mother had the maiden name Rigassi and their grandmother had the last name Mario, she is likely their paternal grandmother. Grandma Mia, a grandmother mentioned by Mario, may or may not be the same character as Grandma Mario. Her 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 enlist 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 succeed, 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 makes her return by smacking Mario and scolding him.
Looking around Mario Brothers Plumbing, Grandma Mario is appalled to see the messy condition the place is in and is unable to describe it in her own words. Luigi suggests that it is a dump, but she instead tells him to shut up and not talk back to her. Grandma Mario then exclaims that their dumpy heart broke her heart and makes her the laughing stock of wherever she had been buried, despite the fact that she is unable to remember the location, which Mario and Luigi guess is either Newark or upper Newark.
Grandma Mario then tells Mario and Luigi to clean up their apartment, tells Luigi to wash behind his ears, and tells Mario to trim his nose hairs, so that she can finally rest easy, even if it has to be in Newark. With that said, Grandma Mario falls back onto a chair as her spirit leaves The Old Psychic Lady's body, presumably returning to her grave.
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 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 with their last name, it is implied that Grandma Mario is respectively the mother and mother-in-law of Papa Mario and Mama Mario, and her husband might have been their grandfather. 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, and Mario Joe and Luigi Bob are either her grandsons or great nephews.