It is clear that in this short story the paralysis that impacts all the characters in Dubliners is the same paralysis that prevents Eveline from escaping the terrible fate that awaits her if she stays in her native Ireland. The family relationships are so much a part of this paralysis, as the reader is told in this short story that she made a promise to her mother before she died that "she would keep the family together as long as she could," and yet note what Eveline thinks of when she reflects on her mother's life and what she lived it for:
As she mused the pitiful vision of her mother's life laid its spell on the very quick of her being--that life of commonplace sacrifices closing in final craziness.
It is this "final craziness" that is summarised in her mother's final words. Eveline, if she stays true to her family, can only expect a similar life of constant sacrifice ending in final pain and death, and yet in the end the pull of her family and the binds that tie her to her kin are stronger than her desire to free herself. A suitable thesis statement to explore Eveline's relationship with her family would therefore be:
In "Eveline," Eveline's relationship with her family is so strong, in spite of her intense desire to escape, that in the end she is unable to free herself from the demands of her family.
Thus thesis statement would allow for a discussion of the conflict that occurs within Eveline and the triumph of the paralysis that affects Eveline, just as much as it affects all other characters in this collection of short stories.