  For a simple graph $G=(V,E)$ and for a pair of vertices $u,v \in V$, we say
that a vertex $w \in V$ resolves $u$ and $v$ if the shortest path from $w$ to
$u$ is of a different length than the shortest path from $w$ to $v$. A set of
vertices ${R \subseteq V}$ is a resolving set if for every pair of vertices $u$
and $v$ in $G$, there exists a vertex $w \in R$ that resolves $u$ and $v$. The
minimum weight resolving set problem is to find a resolving set $M$ for a
weighted graph $G$ such that$\sum_{v \in M} w(v)$ is minimum, where $w(v)$ is
the weight of vertex $v$. In this paper, we explore the possible solutions of
this problem for grid graphs $P_n \square P_m$ where $3\leq n \leq m$. We give
a complete characterisation of solutions whose cardinalities are 2 or 3, and
show that the maximum cardinality of a solution is $2n-2$. We also provide a
characterisation of a class of minimals whose cardinalities range from $4$ to
$2n-2$.
