Like Tiny Tim in Charles Dickens' A Christmas Carol, Bessy Higgins' main role in this book is to just be a sick child who tugs on our heartstrings and gets our sympathy. She knows that she'll never get better from the sickness in her lungs, and no matter how much Margaret Hale tries to cheer her up, she feels like the only time she'll ever know comfort is when she dies.
Margaret, for example, says that she should look forward to springtime, but Bessy has another spring in mind, saying, "I shall have a spring where I'm boun to, and flowers, and amaranths, and shining robes besides" (1.8.32). What she's actually talking about here is dying and going to heaven. Yikes. De-pressing.
Elizabeth Gaskell doesn't let Bessy slip away from us without making a bigger social commentary. It's fairly clear that Bessy has been the victim of poor working conditions and that these conditions have directly contributed to her illness. She's normally not the angry type, but you can feel her frustration come out when she says things like, "All I've been born for is just to work my heart and my life away, and to sicken i' this dree place, wi' them mill-noises in my ears for ever" (1.13.16).
Bessy is certain that there should be more to a human being's life than working, getting sick, and dying. Unfortunately, she'll never live to find out what it is.