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, λ(C)=λ(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.