Step 1: Open appendicectomy uses the grid-iron incision over McBurney's point in the right iliac fossa, splitting the three flat abdominal muscles along the direction of their fibres.
Step 2: The iliohypogastric nerve travels in the neurovascular plane between internal oblique and transversus abdominis at this level, so it lies squarely within the area being divided and is the structure most at risk.
Step 3: If this nerve is cut, the motor supply to the lower internal oblique and transversus is lost, the inguinal canal mechanism is weakened, and a right-sided inguinal hernia may follow.
Step 4: The subcostal nerve (T12) and the T10 and T11 nerves supply skin and muscle at higher abdominal levels and are out of the operative field, leaving the iliohypogastric nerve as the answer.
\[\boxed{\text{Iliohypogastric nerve}}\]