Why do they use graphite electrodes in the Hall–Héroult process?
According to this source, graphite is used as anode material because carbon is easier to oxidize than $\ce{O^2-}$. The anodic oxidation of carbon

$$\ce{C + 2O^2- \rightarrow CO2 + 4e-}$$

proceeds more efficiently than the oxidation of $\ce{O^2-}$ at an inert anode

$$\ce{2O^2- \rightarrow O2 + 4e-}$$

$\ce{C}$ is a much better reducing agent than $\ce{O^2-}$ whose oxidation state is not changed in the reaction with graphite. As a consequence, anodic current density and the deposition rate of aluminium at the cathode will increase.