  Coalescing massive black hole binaries (MBHBs) of $10^{4-7} \rm M_{\odot}$,
forming in the aftermath of galaxy mergers, are primary targets of the space
mission LISA, the {\it Laser Interferometer Space Antenna}. An assessment of
LISA detection prospects requires an estimate of the abundance and properties
of MBHBs that form and evolve during the assembly of cosmic structures. To this
aim, we employ a semi-analytic model to follow the co-evolution of MBHBs within
their host galaxies. We identify three major evolutionary channels driving the
binaries to coalescence: two standard paths along which the binary evolution is
driven by interactions with the stellar and/or gaseous environment, and a novel
channel where MBHB coalescence occurs during the interaction with a third black
hole. For each channel, we follow the orbital evolution of MBHBs with
physically motivated models that include a self-consistent treatment of the
orbital eccentricity. We find that LISA will detect between $\approx 25$ and
$\approx 75$ events per year depending on the seed model. We show that
triple-induced coalescences can range from a few detected events up to $\sim
30\%$ of the total detected mergers. Moreover, even if the standard
gas/stars-driven evolutionary channels should fail and MBHBs were to stall,
triple interactions would still occur as a result of the hierarchical nature of
galaxy formation, resulting in about $\approx 10$ to $\approx 20$ LISA
detections per year. Remarkably, triple interactions among the black holes can
produce coalescing binaries with large eccentricities ($\gtrsim 0.9$) upon
entrance into the LISA band. This eccentricity will remain significant ($\sim
0.1$) also at merger, requiring suitable templates for parameter estimation.
