We present new near-infrared (J and K) magnitudes for 114 RR Lyrae stars in the globular cluster ω Centauri (NGC 5139), which we combine with data from the literature to construct a sample of 180 RR Lyrae stars with J and K mean magnitudes on a common photometric system. This is currently the largest such sample in any stellar system. We also present updated predictions for J- and K-band period-luminosity relations for both fundamental and first-overtone RR Lyrae stars, based on synthetic horizontal branch models with metal abundance ranging from Z=0.0001 to 0.004. By adopting for the ω Cen variables with measured metal abundances an α-element enhancement of a factor of 3 (~0.5 dex) with respect to iron, we find a true distance modulus μ0=13.70+/-0.06+/-0.06 (random and systematic errors, respectively), corresponding to a distance d=5.5+/-0.03+/-0.03 kpc. Our estimate is in excellent agreement with the distance inferred for the eclipsing binary OGLEGC 17, but differ significantly from the recent distance estimates based on cluster dynamics and on high-amplitude δ Scuti stars. Based in part on observations collected with the New Technology Telescope, operated at ESO, La Silla, under programs 066D-0557 and 068D-0545.