Can the family of a deceased family member sue a hospital for negligence?

A spouse can sue for wrongful death. Or her estate (via her personal representative) can sue. But other family cannot sue unless she helped care for them and/or provided their monetary support, since you need to have something beyond an emotional tie to provide grounds to sue. If a surviving spouse, her estate, or family member wants to pursue a case, they should consult with a medical malpractice attorney. 
The other hurdles to a suit, beyond whether someone has sufficient interest in the matter so as to let them sue, are:
1) Did the hospital do anything wrong--even if the hospital was incorrect, if at each stage, their diagnosis and actions were reasonable (even if not right) based on her symptoms, x-rays, other tests, what family reported to her, etc., they did not commit malpractice. Malpractice is more than being medically wrong--it is medical negligence or carelessness, and that generally means not providing care meeting then accepted-standards and/or not doing what the average reasonable doctor would have done in that case.
2) Causation--did the alleged malpractice cause your mother-in-law's death? If given her condition(s), health, etc. she would have suffered the same outcome and passed away even if the had diagnosed her better, then even if the hospital is technically liable, there would be little or no compensation available: compensation is only availabe to the extent the malpractice causes the bad outcome (e.g. death).
1) and 2) are matters to explore with an attorney in detail, if there is anyone eligible to sue who wishes to go that route.
A final consideration: how much might the case be worth? To oversimplify, compensation in wrongful death cases is generally based on a combination of earning potential and remaining life expectance. A suit for the death of, say, a 60-year woman who had many years ahead fo her is very different, and potentially worth much more, than a suit for the death of a 85-year-old who was near the end of her life expectancy. My own mother recently died from sespis from an outpatient surgical procedural; since she was 84 at the time, in only ok health, and had not worked for decades (her only income was social security), we made the decision to not pursue a lawsuit: we could have spent alot of money, and put my father, who was 94, through a great deal of emotional, mental, and physical stress, for comparativel little return.