  The recently discovered gamma-ray burst (GRB) GRB 060218/SN 2006aj is
classified as an X-ray Flash with very long duration driven possibly by a
neutron star. Since GRB 060218 is very near about 140 Mpc and very dim,
one-year observation by Swift suggests that the rate of GRB 060218-like events
might be very high so that such low luminosity GRBs (LL-GRBs) might form a
different population from the cosmological high luminosity GRBs (HL-GRBs). We
found that the high energy neutrino background from LL-GRBs could be comparable
with that from HL-GRBs. If each neutrino event is detected by IceCube, later
optical-infrared follow-up observations such as by Subaru and HST have
possibilities to identify a Type Ibc supernova associated with LL-GRBs, even if
gamma- and X-rays are not observed by Swift. This is in a sense a new window
from neutrino astronomy, which might enable us to confirm the existence of
LL-GRBs and to obtain information about their rate and origin. We also argue
LL-GRBs as high energy gamma-ray and cosmic-ray sources.
