Dynamical formation channels can produce black hole binaries with nonzero eccentricities. Template-based searches by the LIGO-Virgo Collaboration currently do not include templates of such binaries. To detect and characterize these signals one can use algorithms designed for generic gravitational-wave transients. BayesWave is a Bayesian algorithm using sine-Gaussian wavelets for unmodeled reconstructions of signal waveforms and parameters. We present a comprehensive study on how well gravitational-wave signals of binary black holes with nonzero eccentricities are recovered by BayesWave. We used two different waveform models to produce simulated signals of eccentric binary black holes and embedded them in simulated noise of design-sensitivity Advanced LIGO detectors. We studied the network overlaps and point estimates of central moments of signal waveforms recovered by BayesWave as a function of eccentricity.