We investigate the acceleration of electrons and positrons by magnetic-field-aligned electric fields in the polar funnel of an accreting black hole (BH). Applying the pulsar outer-gap theory to BH magnetospheres, we find that such a lepton accelerator arises in the immediate vicinity of the event horizon due to frame-dragging, and that their gamma-ray luminosity increases with decreasing accretion rate. Furthermore, we demonstrate that the gamma-ray flux is enhanced along the rotation axis by more than an order of magnitude if the BH spin increases from a=0.90M to a = 0.9999M. As a result, if a ten-solar-mass, almost-maximally rotating BH is located within 3 kpc, when its accretion rate is between 0.005 and 0.01 per cent of the Eddington rate, its high-energy flare becomes detectable with the Fermi/Large Area Telescope, provided that the flare lasts longer than 1.2 months and that we view the source nearly along the rotation axis. In addition, its very high energy flux is marginally detectable with the Cherenkov Telescope Array, provided that the flare lasts longer than a night and that our viewing angle is about 45 deg with respect to the rotation axis.