Context. Observations of auroral emissions are powerful means to remotely sense the space plasma environment around planetary bodies and ultracool dwarfs. Therefore successful searches and characterization of aurorae outside the Solar System will open new avenues in the area of extrasolar space physics. Aims. We aim to demonstrate that brown dwarfs are ideal objects to search for UV aurora outside the Solar System. We specifically search for UV aurora on the late-type T6.5 brown dwarf 2MASS J12373919+6526148 (in the following 2MASS J1237+6526). Methods. Introducing a parameter referred to as auroral power potential, we derive scaling models for auroral powers for rotationally driven aurora applicable to a broad range of wavelengths. We also analyze Hubble Space Telescope observations obtained with the STIS camera at near-UV, far-UV, and Ly-alpha wavelengths of 2MASS J1237+6526. Results. We show that brown dwarfs, due to their typically strong surface magnetic fields and fast rotation, can produce auroral UV powers on the order of 10(19) watt or more. Considering their negligible thermal UV emission, their potentially powerful auroral emissions make brown dwarfs ideal candidates for detecting extrasolar aurorae. We find possible emission from 2MASS J1237+6526, but cannot conclusively attribute it to the brown dwarf due to low signal-to-noise values in combination with nonsystematic trends in the background fluxes. The observations provide upper limits for the emission at various UV wavelength bands. The upper limits for the emission correspond to a UV luminosity of similar to 1 x 10(19) watt, which lies in the range of the theoretically expected values. Conclusions. The possible auroral emission from the dwarf could be produced by a close-in companion and/or magnetospheric transport processes.