The mini-broad absorption line (BAL) quasar RX J0911.4+0551 was observed with the Advanced CCD Imaging Spectrometer of the Chandra X-Ray Observatory for ~29 ks as part of a gravitational lens (GL) survey aimed at measuring time delays. Timing analysis of the light curve of the lensed image A2 shows a rapid flux variation with a duration of about 2000 s. A Kolmogorov-Smirnov test shows that the probability that a constant intensity source would produce the observed variability is less than ~0.2%. We discuss possible origins for the observed short-term X-ray variability. Our gravitational lens models for the RX J0911.4+0551 GL system predict a time delay of less than a day between images A1 and A2. The rapid variability combined with the predicted short time delay make RX J0911.4+0551 an ideal system to apply the GL method for estimating the Hubble constant. We describe the prospects of measuring H0 within single X-ray observations of GL systems with relatively short time delays. Modeling of the spectrum of the mini-BAL quasar RX J0911.4+0551 suggests the presence of an intrinsic absorber. Partial covering models are slightly preferred over models that contain absorption due to intrinsic ionized or neutral gas.