  The recently investigated $1T$-polymorph of monolayer NbSe$_2$ revealed an
insulating behaviour suggesting a star-of-David phase with
$\sqrt{13}\,\times\sqrt{13}$ periodicity associated with a Mott insulator,
reminiscent of $1T$-TaS$_2$. In this work, we examine this novel
two-dimensional material from first principles. We find an instability towards
the formation of an incommensurate charge-density-wave (CDW) and establish the
star-of-David phase as the most stable commensurate CDW. The mottness in the
star-of-David phase is confirmed and studied at various levels of theory: the
spin-polarized generalized gradient approximation (GGA) and its extension
involving the on-site Coulomb repulsion (GGA+$U$), as well as the dynamical
mean-field theory (DMFT). Finally, we estimate Heisenberg exchange couplings in
this material and find a weak nearest-neighbour ferromagnetic coupling, at odds
with most Mott insulators. We point out the close resemblance between this
star-of-David phase and flat-band ferromagnetism models.
