  Let $\Phi$ be a concave function on $(0,\infty)$ of strictly lower type
$p_{\Phi}\in(0,1]$ and $\omega\in
A^{\mathop\mathrm{loc}}_{\infty}(\mathbb{R}^n)$. We introduce the weighted
local Orlicz-Hardy space $h^{\Phi}_{\omega}(\mathbb{R}^n)$ via the local grand
maximal function. Let $\rho(t)\equiv t^{-1}/\Phi^{-1}(t^{-1})$ for all
$t\in(0,\infty)$. We also introduce the $\mathop\mathrm{BMO}$-type space
$\mathop\mathrm{bmo}_{\rho,\,\omega}(\mathbb{R}^n)$ and establish the duality
between $h^{\Phi}_{\omega}(\mathbb{R}^n)$ and
$\mathop\mathrm{bmo}_{\rho,\,\omega}(\mathbb{R}^n)$. Several real-varaiable
characterizations of $h^{\Phi}_{\omega}(\mathbb{R}^n)$ are presented. Using the
atomic characterization, we prove the existence of finite atomic decompositions
achieving the norm in some dense subspaces of
$h^{\Phi}_{\omega}(\mathbb{R}^n)$. As applications, we show that the local
Riesz transforms are bounded on $h^{\Phi}_{\omega}(\mathbb{R}^n)$, the local
fractional integrals are bounded from
{\normalsize$h^p_{\omega^p}(\mathbb{R}^n)$} to
{\normalsize$L^q_{\omega^q}(\mathbb{R}^n)$} when $q>1$ and from
{\normalsize$h^p_{\omega^p}(\mathbb{R}^n)$} to
{\normalsize$h^q_{\omega^q}(\mathbb{R}^n)$} when $q\le 1$, and some
pseudo-differential operators are also bounded on both
$h^{\Phi}_{\omega}(\mathbb{R}^n)$. All results for any general $\Phi$ even when
$\omega\equiv 1$ are new.
