The Lizzie Borden case, much like the O.J. Simpson case, remains unsolved because no one was ever convicted of the crime, and Borden was the most likely, indeed the only suspect. Although there was overwhelming evidence of Borden's guilt (she had previously attempted to purchase prussic acid from a druggist for a spurious reason; she was the only person with access to the house other than those murdered, etc.) she was found not guilty. Borden's lawyer was a former governor of Massachusetts who managed to win sympathy for her with the all-male jury. Under the Fifth Amendment's protection against double jeopardy, Borden could not be tried again. Much as in the O.J. Simpson case, prosecutors decided not to pursue other leads as they believed that the guilty party simply got away with the crime. Borden moved away after the trial, but could not avoid the court of public opinion. She never married, and ultimately died lonely and alone.
The bitter irony is that if Borden had been found guilty and punished for the axe murder, the famous rhyme about her would never have become popular.