We have discovered a ˜420-d modulation, with associated X-ray dips, in Rossi X-Ray Timing Explorer-All Sky Monitor/Monitor of All-Sky X-ray Image/Swift-Burst Alert Telescope archival light curves of the short-period (3.2 h) black hole X-ray transient, Swift J1753.5-0127. This modulation only appeared at the end of a gradual re-brightening, approximately 3 yr after the initial X-ray outburst in mid-2005. The same periodicity is present in both the 2-20 and 15-50 keV bands, but with a ˜0.1 phase offset (≈40 d). Contemporaneous photometry in the optical and near-infrared reveals a weaker modulation, but consistent with the X-ray period. There are two substantial X-ray dips (very strong in the 15-50 keV band, weaker at lower energies) that are separated by an interval equal to the X-ray period. This likely indicates two physically separated emitting regions for the hard X-ray and lower energy emission. We interpret this periodicity as a property of the accretion disc, most likely a long-term precession, where the disc edge structure and X-ray irradiation are responsible for the hard X-ray dips and modulation, although we discuss other possible explanations, including Lense-Thirring precession in the inner disc region and spectral state variations. Such precession indicates a very high mass ratio low-mass X-ray binary, which even for a ˜10 M⊙ BH requires a brown dwarf donor (˜0.02 M⊙), making Swift J1753.5-0127 a possible analogue of millisecond X-ray pulsars. We compare the properties of Swift J1753.5-0127 with other recently discovered short-period transients, which are now forming a separate population of high-latitude BH transients located in the galactic halo.