I'm not a coder myself but I have been pointed in the right direction and it's a really easy change!
In CvUnit.cpp, look for void CvUnit::init
or rather directly in that section for
iUnitName = GC.getGameINLINE().getUnitCreatedCount(getUnitType());
4 lines below you will find this code to be changed:
int iOffset = GC.getGameINLINE().getSorenRandNum(iNumNames, "Unit name selection");
comment it out and replace it by:
int iOffset = 0;
and you're done!
Note that it's not elegant coding but, like this, it is very easy to change it without messing up with the code. A small trick for amateurs like us!