in a drop down list for the selected value is saved to the parameter in the lambda expression. You have your view model there. What you need to do is add a selected parameter to your view model
public string Selected { get; set; }
then on your view change your lambda to look at that value
@Html.DropDownListFor(h => h.Selected, new SelectList(Model.MuseSites, "Key", "Description"), "Select one")
on your get you can preset the drop down by setting Selected and the selected value will be saved to that field