Step 1: Map the three nephric systems in time order: pronephros (rudimentary, regresses), mesonephros (interim kidney with the Wolffian duct), and metanephros (the definitive adult kidney). The duct of the mesonephros persists and is the key to this question.
Step 2: Near where the mesonephric duct meets the cloaca, it sprouts a diverticulum. This outgrowth is the ureteric bud, so its parent is the mesonephric duct (mesonephros). The bud later branches to build the collecting system - ureter, pelvis, major and minor calyces, and collecting tubules.
Step 3: Knock out the rest. The metanephric blastema, derived from the metanephros, contributes nephrons (glomeruli, Bowman capsule, proximal and distal tubules, loop of Henle) but not the bud itself. The pronephros never persists to form anything. The genital sinus builds the bladder and urethra, unrelated to the ureteric bud.
\[\boxed{\text{Mesonephros (mesonephric duct)}}\]