Hello,
the solution would depend on the individual interface where you want to have a different ordering and the amount of changes you are willing to do. There is no single solution - apart from renaming the sites / folders - that will get you a consistent, different ordering across all interfaces without significant changes to the standard Alfresco.
For Share you could e.g. start by adapting the different Repository data web scripts that fetch sites and file/folders, so that it uses a different property for ordering (this could be a new property you add to the data model which may simply hold the ordering index).
Again, this is not a simple thing to do and I doubt we could list all the potential changes you would have to do to get this working across all interfaces. Is it really worth the effort? Apart from pure preference, is there a real business case for this requirement that is important to you / your client?
Regards
Axel