Step 1: Recall that the uncertainty principle limits how finely a state can be localized in the position-momentum plane: the product $\sigma_x\sigma_p$ cannot be smaller than $\hbar/2$.
Step 2: The equality $\sigma_x\sigma_p=\hbar/2$ is achieved only by a Gaussian wave packet.
Step 3: For a harmonic oscillator the ground-state wave function is precisely such a Gaussian, so it is the minimum-uncertainty state and occupies the smallest possible cell.
Step 4: Note the distractor $\hbar\omega/2$ is the ground-state energy, not a phase-space area, so it is dimensionally wrong here.
Step 5: Therefore the minimum phase-space volume is
\[\boxed{\dfrac{\hbar}{2}}\]