  We investigate the ground-state and the finite-temperature properties of the
bond-random $s=1/2$ Heisenberg model on a square lattice with frustrating
nearest- and next-nearest-neighbor antiferromagnetic interactions, $J_1$ and
$J_2$, by the exact diagonalization and the Hams--de Raedt methods. The
ground-state phase diagram of the model is constructed in the randomness versus
the frustration ($J_2/J_1$) plane, with the aim of clarifying the effects of
randomness and frustration in stabilizing a variety of phases. We find that the
randomness induces the gapless quantum spin liquid (QSL)-like state, the
random-singlet state, together with the spin-glass state in a certain range of
parameter space. The spin-glass state might be stabilized by employing the
lattice directional degrees of freedom associated with the stripe-type magnetic
order of the regular model. Possible implications to recent experiments on the
square-lattice mixed-crystal antiferromagnet Sr$_2$Cu(Te$_{0.5}$W$_{0.5}$)O$_6$
exhibiting the gapless QSL-like behaviors are discussed.
