You have to change the name of the Layout. I used this snippet below in a button to change the Layout name, which in turn is then used as the default file name. Just make sure that you don't use invalid file name characters (like ':').
protected override async void OnClick()
{
Map map = null;
await QueuedTask.Run(() =>
{
var layoutItem = Project.Current.GetItems<LayoutProjectItem>().FirstOrDefault(i => i.Name.Equals ("layout", StringComparison.OrdinalIgnoreCase));
if (layoutItem != null)
{
var layout = layoutItem.GetLayout();
var defLayout = layout.GetDefinition();
var now = DateTime.Now.TimeOfDay;
defLayout.Name = $@"msecs - {now.TotalMilliseconds}";
layout.SetDefinition(defLayout);
}
});
}