  Employing the density-functional theory with local density approximation, we
show that the fully hydrogenated monolayer-hexagonal boron nitride (H$_2$BN)
has a direct-band gap of 2.96 eV in the blue-light region while the pristine
$\textit{h}$-BN has a wider indirect-band gap of 4.78 eV. The hole-doped
H$_2$BN is stable at low carrier density ($n$) but becomes dynamically unstable
at higher $n$. We predict that it is a phonon-mediated superconductor with a
transition temperature ($T_c$) which can reach $\sim$31 K at $n$ of $1.5\times$
$10^{14}$ holes cm$^{-2}$ near the lattice instability. The $T_c$ could be
enhanced up to $\sim$82 K by applying a biaxial tensile strain at 6 % along
with doping at $n$ of $3.4\times$ 10$^{14}$ holes cm$^{-2}$ close to a new
lattice instability.
