  Gravitational wave (GW) astronomy has revolutionized our capacity to explore
nature. The next generation of observatories, among which the space-borne
detector Laser Interferometer Space Antenna LISA, is expected to yield orders
of magnitude of signal-to-noise ratio improvement, and reach fainter and novel
features of General Relativity. Among them, an exciting possibility is the
detection of GW memory. Interpreted as a permanent deformation of the
background spacetime after a GW perturbation has passed through the detector,
GW memory offers a novel avenue to proof-test General Relativity, access the
non-linear nature of gravity, and provide complementary information for a
better characterization of the GW source. Previous studies have shown that GW
memory detection from individual mergers of massive black hole binaries is
expected with LISA. However, these works have not simulated the proper time
domain response of the detector to the GW memory. This work is filling this gap
and presents the detection prospects of LISA regarding GW memory and the
expected signature of GW memory on the data-streams using the most up-to-date
LISA consortium simulations of the response, as well as GW memory time-series
computation inherited from numerical relativity. We will confront the LISA
observation window to massive black hole binary mergers' population forecasted
with the state-of-the-art population models and evaluate the odds and the
expected accuracies regarding GW memory observations in the LISA lifetime. We
conclude that GW memory will be a key feature of several events detected by
LISA, and will help to exploit the scientific potential of the mission fully.
