Billy Holiday (or "Lady Day," as many called her) struggled with addiction throughout her too-brief but stellar career. Although she only lived to age 44, her work during her relatively short time on earth was forever memorable. If the reviews of her final performances in the mid- to late 1950s are any indication, her failing health and troubles with alcohol and drugs did not diminish her incredible talent.
Billie Holiday's untimely passing came at a turning point of her career. Had she survived and overcome her drug and alcohol addiction, her career would have continued to grow over time, and today's audiences would be more familiar with her body of work as a result of the increased exposure.
Many famous people who died at a young age, for whatever reason, are still remembered today. Young people today are well aware of musicians like: Elvis Presley, Jim Morrison, Janis Joplin, etc. All of these musicians died well before their time and yet, are revered by our youth. As long as their music remains accessible, their fame will live on.
Ultimately, some people are more famous because of their untimely death. Marilyn Monroe is another example of this. In Holiday's case, people remember her for what she accomplished in her early life, rather than what she might have done when she was elderly. This kept her famous and celebrated, even though she died younger than she would have otherwise.