  The particle discovered in the Higgs boson searches at the LHC with a mass of
about 125 GeV is compatible within the present uncertainties with the Higgs
boson predicted in the Standard Model (SM), but it could also be identified
with one of the neutral Higgs bosons in a variety of Beyond the SM (BSM)
theories with an extended Higgs sector. The possibility that an additional
Higgs boson (or even more than one) could be lighter than the state that has
been detected at 125 GeV occurs generically in many BSM models and has some
support from slight excesses that were observed above the background
expectations in Higgs searches at LEP and at the LHC. The couplings between
additional Higgs fields and the electroweak gauge bosons in BSM theories could
be probed by model-independent Higgs searches at lepton colliders. We present a
generator-level extrapolation of the limits obtained at LEP to the case of a
future $e^+e^-$ collider, both for the search where the light Higgs boson
decays into a pair of bottom quarks and for the decay-mode-independent search
utilising the recoil method. We find that at the ILC with a c.m. energy of 250
GeV, an integrated luminosity of 500 fb^{-1} and polarised beams, the
sensitivity to a light Higgs boson with reduced couplings to gauge bosons is
improved by more than an order of magnitude compared to the LEP limits and goes
much beyond the projected indirect sensitivity of the HL-LHC with 3000 fb^{-1}
from the rate measurements of the detected state at 125 GeV.
