A broken bud is a symbol for a premature or untimely death—a life that has ended before it has bloomed.
Most often, the broken bud (most I have seen are roses) is found on the gravestone a child or young adult.
By the way, there is a particularly interesting example of a broken bud posted on Gravely Speaking, a blog by Douglas M. Rife.
Lorain M. Andrews, Mill Creek Cemetery, Delaware County, Ohio
Anna M. Keys, Maple Grove Cemetery, Franklin County, Ohio
Sarah E. Day, Houck Cemetery, Knox County, Ohio
Ada Kennedy, Sunbury Memorial Park, Delaware County, Ohio
Rosa E. and Henrietta H. Adams, Glendale Cemetery, Morrow County, Ohio