We have a couple of projects where we wanted to gain more control over the sort order of some of the default Umbraco tabs like "Properties" or "Child items". Although I don't know of a way to do this through the Umbraco backoffice or with configuration, it is suprisingly easy to write some custom code to handle this.
The idea is pretty simple. The Umbraco core (at least as of Umbraco 7.2.x) uses AutoMapper to map the IContent into ContentItemDisplay objects. Using AutoMapper, we can interecept these mappings and tack on some additional adjustments.
Then we just need to make sure that on ApplicationStarted, we actually add the new mapping adjustments.