Does this graph problem have a formal name? Given an undirected weighted graph where an edge exists between every pair of nodes (n1,n2) with cost C(n1,n2), find the shortest path (possibly revisiting nodes, possibly revisiting edges) through the graph that visits each node.  Algo can pick any source node it wants.  No need for the path to terminate back at the source node.  Path cost is cumulative over all edge visits, so, if you traverse an edge with cost 5 and then traverse it again later it costs you 10 (i.e. re-traversing an already explored edge isn't free!)
The TSP problem, given a graph and an edge-weight function, asks to find a tour visiting each vertex exactly once having minimum length.

I think your problem is equivalent to the Metric TSP problem, that is the TSP problem on the complete graph where the weight of each edge $uv$ is the length of a path between $u$ and $v$ with minimum weight in your original graph.

Indeed, a minimum walk on your original graph corresponds to a minimum tour in the newly defined clique, and vice-versa.

For the first side, consider a minimum walk in the original graph. If some node is repeated (say $v$ is repeated: $\ldots, v, \ldots, u, v, w, \ldots$) one can get rid of the repetition by replacing the portion $u,v,w$ by $u,w$ in the shortest path graph. Doing this for each repetition one gets a tour in the shortest path graph having the same length as the original walk. Since the walk was optimal the tour is optimal too.

For the other side, given an optimal tour of the new graph, replace each edge $uv$ of the tour by a shortest weighted path from $u$ to $v$ is the original graph. This walk may contain some repetitions, but has now the same length as the tour, and it is optimal.