  Density functional theory calculations have been used to identify stable
layered Li-$M$-B crystal structure phases derived from a recently proposed
binary metal-sandwich (MS) lithium monoboride superconductor. We show that the
MS lithium monoboride gains in stability when alloyed with electron-rich metal
diborides; the resulting ordered Li$_{2(1-x)}M_x$B$_2$ ternary phases may form
under normal synthesis conditions in a wide concentration range of $x$ for a
number of group-III-V metals $M$. In an effort to pre-select compounds with the
strongest electron-phonon coupling we examine the softening of the in-plane
boron phonon mode at $\Gamma$ in a large class of metal borides. Our results
reveal interesting general trends for the frequency of the in-plane boron
phonon modes as a function of the boron-boron bond length and the valence of
the metal. One of the candidates with a promise to be an MgB$_2$-type
superconductor, Li$_2$AlB$_4$, has been examined in more detail: according to
our {\it ab initio} calculations of the phonon dispersion and the
electron-phonon coupling $\lambda$, the compound should have a critical
temperature of $\sim4$ K.
