  We study the dynamics of a solar-type star orbiting around a black hole
binary (BHB) in a nearly coplanar system. We present a novel effect that can
prompt a growth and significant oscillations of the eccentricity of the stellar
orbit when the system encounters an "apsidal precession resonance", where the
apsidal precession rate of the outer stellar orbit matches that of the inner
BHB. The eccentricity excitation requires the inner binary to have a non-zero
eccentricity and unequal masses, and can be created even in non-coplanar
triples. We show that the secular variability of the stellar orbit's apocenter,
induced by the changing eccentricity, could be potentially detectable by
\textit{Gaia}. Detection is favorable for BHBs emitting gravitational waves in
the frequency band of the Laser Interferometer Space Antenna (LISA), hence
providing a distinctive, multi-messenger probe on the existence of stellar-mass
BHBs in the Milky Way.
