By definition, am aristocrat is someone who belongs to the upper class of society. Miss Havisham is definitely rich. She lives in a huge manor, Satis House, and has land and a gate surrounding it. Miss Havisham is important, and she knows it.
However, it is not just having money that makes someone an aristocrat. Despite her peculiarities, Miss Havisham is well-respected in the community. Uncle Pumblechook and Mrs. Joe cannot wait to have Pip visit her.
I had heard of Miss Havisham up town—everybody for miles round had heard of Miss Havisham up town—asan immensely rich and grim lady who lived in a large and dismal house barricaded against robbers, and wholed a life of seclusion. (ch 7, enotes etext p. 36)
They think an association with her raises them in the community, and may benefit them monetarily. Miss Havisham is also an aristocrat because she expects people to bow to her every whim, and do whatever she says. They do! Pip succumbs to her crazy demands because he has to, because not pleasing such an important woman is not an option.
Miss Havisham is the way she is because she was jilted on her wedding day. To someone who expects everything to go her way, this was a huge blow from which she never recovered. So becoming reclusive and staying in a wedding dress for decades is actually somewhat aristocratic behavior too.