Coherent electron cooling is a novel method to cool dense hadron beams on timescales of a few hours. This method uses a copropagating beam of electrons to pick up the density fluctuations within the hadron beam in one straight section and then provides corrective energy kicks to the hadrons in a downstream straight, cooling the beam. Microbunched electron cooling is an extension of this idea, which induces a microbunching instability in the electron beam as it travels between the two straights, amplifying the signal. However, initial noise in the electron bunch will also be amplified, providing random kicks to the hadrons downstream which tend to increase their emittance. In this paper, we develop an analytic estimate of the effect of the electron noise and benchmark it against simulations. We also discuss how this effect has impacted the cooler design. Published by the American Physical Society 2024