Is this a known problem in graph theory?
If $G=(V,E)$, with $V=\{v_1,v_2,...,v_n\}$ and weights $\{c_{i,j}, i=1,2,...,n, j=1,2,...,p\}$ is the given graph, then we can construct the strong product (I finally found the name of the operation) $G\boxtimes K_p$ of $G$ and $K_p$, where $K_p$ is the complete graph with $p$ vertices. This is the graph with vertices $\{v_{i,j},i=1,2,...,n, j=1,2,...,p\}$ and edges $\{v_{a,b},v_{c,d}\}$ where either:

$a=c$,
$b=d$ or
$\{v_a,v_c\}\in E$. (The actual condition of the strong product reduces to this since in $K_p$ all vertices are adjacent).

We give the vertex $v_{i,j}$ the weight $c_{i,j}$, for $i=1,2,...,n$ and $j=1,2,...,p$.
The problem on $G$ is equivalent to the problem minimum (maximum) weighted independent set in the weighted $G\boxtimes K_p$. If a vertex $v_{i,j}$ of the new graph is chosen this corresponds to choosing vertex $v_i$  of the original graph and using the $j$-th weight $c_{i,j}$ corresponding to it.
The set of edges of $G\boxtimes K_p$ are exactly those that prevent the corresponding choices in $G$ to use adjacent vertices or reusing weights with the same index:

Condition $1$ defines edges in the strong product that prevent the equivalent of using two weights from the same original vertex.
Condition $2$ prevents using the weights with the same index from different vertices of the original graph.
Condition $3$ prevents that two vertices that were neighbors in the original graph are selected.

Example:
If $G$ is the graph

and $p=2$, then $G\boxtimes K_2$ would be the graph

Images created with this tool.