  Motivated by the recently reported evidence of an association between a
high-energy neutrino and a gamma-ray flare from the blazar TXS 0506+056, we
calculate the expected high-energy neutrino signal from past, individual
flares, from twelve blazars, selected in declinations favourable for detection
with IceCube. To keep the number of free parameters to a minimum, we mainly
focus on BL Lac objects and assume the synchrotron self-Compton mechanism
produces the bulk of the high-energy emission. We consider a broad range of the
allowed parameter space for the efficiency of proton acceleration, the proton
content of BL Lac jets, and the presence of external photon fields. To model
the expected neutrino fluence we use simultaneous multi-wavelength
observations. We find that in the absence of external photon fields and with
jet proton luminosity normalised to match the observed production rate of
ultra-high-energy cosmic rays, individual flaring sources produce a modest
neutrino flux in IceCube, $\lesssim10^{-3}$ muon neutrinos with energy
exceeding 100 TeV, stacking ten years of flare periods selected in the >800 MeV
Fermi energy range from each source. Under optimistic assumptions about the jet
proton luminosity and in the presence of external photon fields, we find that
the two most powerful sources in our sample, AO 0235+164, and OJ 287, would
produce, in total, $\approx 3$ muon neutrinos during ten years of Fermi flaring
periods, in future neutrino detectors with total instrumented volume $\sim$ten
times larger than IceCube,or otherwise, constrain the proton luminosity of
blazar jets.
