  We present first-principles results on the magnetoelastic coupling in
$\alpha$-RuCl$_3$ and uncover a striking dependence of the magnetic coupling
constants on strain effects. Different magnetic interactions are found to
respond very unequally to variations in the lattice, with the Kitaev
interaction being the most sensitive. Exact diagonalization results on our
magnetoelastic model reproduce recent measurements of the structural
Gr\"uneisen parameter and explain the origin of the negative magnetostriction
of $\alpha$-RuCl$_3$, disentangling contributions related to different
anisotropic interactions and g factors. Uniaxial strain perpendicular to the
honeycomb planes is predicted to reorganize the relative coupling strengths,
strongly enhancing the Kitaev interaction while simultaneously weakening the
other anisotropic exchanges under compression. Uniaxial strain may therefore
pose a fruitful route to experimentally tune $\alpha$-RuCl$_3$ nearer to the
Kitaev limit.
