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Simply put, Miss Havisham sees through her relatives' sickeningly sweet facades. She knows that they come to visit and pay her compliments only to get her inheritance. While she enjoys manipulating them, she has no intention of willing them anything of value.
Similarly, her own brother set her up for Compeyson's jilting--obviously, the most significant event in her life--and she quite possibly distrusts anyone related to her because of his actions.
Miss Havisham is a very proud and arrogant lady. Being the only child of a rich country gentleman she had always been used to having all her whims and fancies catered to and she had always treated all her relations with disrespect. In Ch. 22 Herbert tells Pip about Miss Havisham's origins:
Miss Havisham, you must know, was a spoilt child. Her mother died when she was a baby, and her father denied her nothing. Her father was a country gentleman down in your part of the world, and was a brewer.
One of the important reasons that she disliked her relations was that she was cheated by her step brother Arthur who introduced her to Compeyson with the idea of swindling all her property and money.
In Ch.11 Pip recognises that all of Miss Havisham's relations are 'toadies' and 'humbugs.'
they were all toadies and humbugs, but that each of them pretended not to know that the others were toadies and humbugs: because the admission that he or she did know it, would have made him or her out to be a toady and humbug.
So, if a small boy is able to recognize these strangers on his first meeting with them as hypocrites how obvious it would have been for Miss Havisham herself to recognize the fact that her relations are visiting her on her birthday just to pretend that they love her. She knows fully well that they are waiting for her to die so that they can share her property and wealth.
In Ch.57 we learn of the exact details of the will of Miss Havisham. She disappointed all her sycophantic relations by leaving behind very measly sums for each of them. However, she rewarded Matthew Pocket who never visited her by bequeathing him a sum of four thousand pounds! The bitter lessons she had learnt had finally made her an astute person.
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