Dora's mother is, um, Dora's mother. Truth is, we spend so little time with her that we can't even refer to her by a proper name. Guess we could call her Joshua's grandmother, but that's just kicking the name can down the generational road, isn't it?
While we don't spend a lot of time with her or get a proper name, we do get a sense of her character. Dora and her mother don't get along, as evidenced by their little kerfuffle before the engagement party.
The reason for the argument? Dora's mother seems to want things for her daughter that Dora doesn't. In this case, Dora's a free-spirited girl, but her mother wants her to settle down. It's an old story.
During this argument, Dora's mother says, "If you don't get up immediately, I swear on your father's deathbed I'll never speak to you again. For the rest of my life!" One gets the sense that Dora's mother is the driving force behind Dora's marriage to Rodolfo.
Also, judging by the lavish house her late husband left her, we can surmise that Dora's mother is a socialite. This leads us to wonder if her desire to see Dora marry Rodolfo isn't part of a plan to keep Dora and the family connected to Italian high society.
During the film's second half, we learn that Dora's mother's threat almost came to pass. After Dora runs off with Guido, she and her mother become estranged for years—as evidenced by the fact that Joshua says he's never met his grandmother. But Dora's mother hatches a plan to get back into her daughter's life.
Grandchildren will do that to you.
When Guido's taken to the Prefect's house, he leaves Joshua to mind the bookstore, and Dora's mother comes in pretending to be a customer. She talks to Joshua for the first time, asking what he wants for his birthday and complimenting him on how clever he is. She then asks him to deliver a letter to his mother (at this point, Joshua, who really is a clever boy, figures out who she is).
We don't know what was written in that letter, but it must have been dynamite. Soon after, Dora and her mother are driving to Joshua's birthday party and clearly have made up.
Unfortunately, before Dora's mother can be properly introduced to her new family, she and Dora discover that Joshua, Guido, and Eliseo have been taken by the government and are to be sent to the concentration camp. That's the last we see of her.
We're betting she wishes she'd have come a few years sooner.