The minimum variance (MV) beamformer is a method that has the potential to enhance the resolution and contrast of ultrasound images. However, it suffers from sensitivity to speed of sound errors and aberration. Although there have been several studies on the application of phase aberration correction (PAC) methods to conventional delay-and-sum (DAS) beamforming, the benefits of PAC methods in mitigating the effects of phase aberration in MV beamformed images are not well understood. A study of this type would be helpful in designing a robust beamformer based on PAC knowledge present in the literature. This study analyzes three PAC algorithms (multi-lag cross-correlation, Rigby's beamsum and scaled covariance matrix) and their impact on the performance of the MV beamformer in the presence of second order phase aberrations. The PAC methods in combination with the MV beamformer were tested on simulated and experimental data corrupted with an electronically created near field phase aberrator. It is shown that all gains in performance of the MV beamformer with respect to DAS is lost at high aberration strengths. For instance, at 60 ns of aberration the lateral resolution of DAS degrades by 22% while MV degrades by 600%. It is also shown that basic PAC methods improve the aberrated MV beamformer. PAC methods reduces degradation in lateral resolution from 600% to 5%. Similar improvements are observed in peak sidelobe level (96% to 27%), contrast (88% to 49% for the simulations and 43% to 15% for experiments) and contrast-to-noise ratio (86% to 42% for the simulations and 68% to 55% for experiments). These enhancements allow the MV beamformer to outperform DAS even in the strongest aberration case.