Step 1: What the question wants.
We need the brain region that sets the basic rhythm of breathing, not just modifies it.
Step 2: Rule out the distractors.
The cerebrum handles voluntary control of breathing (like holding your breath), the hypothalamus manages body temperature and thirst, and the pons only fine tunes the rhythm through the pneumotaxic and apneustic centres. None of these generate the basic pattern on their own.
Step 3: Locate the actual rhythm generator.
Deep in the medulla oblongata sit the dorsal and ventral respiratory groups of neurons. These fire in a repeating cycle and are the real pacemaker of breathing, adjusting rate based on $CO_2$ and $H^+$ levels in the blood.
Final answer: Option 2, Medulla oblongata.