We study aging during surface growth processes described by the one-dimensional Kardar-Parisi-Zhang equation. Starting from a flat initial state, the systems undergo simple aging in both correlators and linear responses, and its dynamical scaling is characterized by the aging exponents a = -1/3, b = -2/3, lambda(C) = lambda(R) = 1, and z = 3/2. The form of the autoresponse scaling function is well described by the recently constructed logarithmic extension of local scale invariance.