Bud is the genetic clone of Super Bunny, created when the hero touched a large pod on a Martian Flower. The pod was a genetic template, which took DNA of whoever touched it and attempted to recreate it as a clone. However, since Super Bunny's DNA is so advanced, the primitive plant could only create a baby version with limited powers and limited IQ. Nevertheless, Bud is still a valued member of Super Bunny's crime-fighting team and is loved by his 'father'.
More importantly, he is the next step in Luke Smith's work in solving the riddle of Super Bunny's existance. Since no DNA sample of Super Bunny can be taken, how did the flower recreate him? It leaves many questions unanswered.