We have a couple of projects that we wanted to do this on as well. Unfortunately, I can't claim credit for getting this figured out. I just looked around in the source until I figured out how we did it. :)

It looks like in the core of umbraco, umbraco uses AutoMapper to map the IContent into these ContentItemDisplay objects. We can use AutoMapper to intercept these mappings and make additional adjustments. In our case, we are rearranging the tabs.

internal class TabMover
{
    internal static void BindMappings()
    {
        Mapper.CreateMap<IContent, ContentItemDisplay>().AfterMap(HandleAfterMap);
    }

    internal static void HandleAfterMap(IContent content, ContentItemDisplay contentItem)
    {
        ReorderTabs(contentItem);
    }

    private static void ReorderTabs(ContentItemDisplay contentItem)
    {
        if (contentItem.Tabs.Any(t => t.Label == "Child items"))
        {
            var tabs = new List<Tab<ContentPropertyDisplay>>();
            tabs.AddRange(contentItem.Tabs);

            var list = tabs.OrderBy(x => (x.Label == "Child items" ? 2 : 1 )); 

            contentItem.Tabs = list;
        }
    }
}

Then you just make sure that on ApplicationStarted, we actually have the TabMover adjust the tab order.

protected override void ApplicationStarted(UmbracoApplicationBase umbracoApplication, ApplicationContext applicationContext)
{
    base.ApplicationStarted(umbracoApplication, applicationContext);

    try
    {
        TabMover.BindMappings();
    }
    catch (Exception ex)
    {                
        Log.Error("BindMappings Failed", ex);
    }
}