Their names referred to Prince Eugene of Savoy (1663-1736), a general within the Imperial Army of the Holy Roman Empire. Eugene grew up in Paris as part of the royal court around King Louis XIV, but because of his poor physique and personal differences involving Eugene, his mother, and the King, Louis wouldn't allow him to enter the French military. Eugene travelled east instead and became an officer in service of the Habsburgs. He came at a perilous time: A massive Ottoman host had been besieging Vienna for a while, and the Emperor had fled to Passau and made camp there. Only a few weeks after his arrival at the Imperial court, Eugene already saw action at the Battle of Vienna, when the Ottomans were successfully driven off.

During the following years of protracted warfare against the Ottomans in the regions south and east of Vienna (much of it took place in what is Serbia today), Eugene quickly proved himself highly adept at being a military commander. He quickly rose through the ranks and finally got his first independent command in 1697, when he was made supreme commander of the Imperial Army. This sounds more pompous than it actually was, however; the Imperial forces were a hodgepodge of troops from all over the Empire that were sent by the many various princes and territories at the call of the Emperor. They had no common hierarchy, structure or drill, and the local rulers often wouldn't exactly send their best. Eugene still managed to get "his" army into shape and got to prove himself and his soldiers' valour at the Battle of Zenta in autumn 1697, where they managed to beat the Ottomans so decisively that it effectively ended the war shortly afterwards.

Eugene fought in so many campaigns and conflicts after that (some of them even against his own native country of France) that I don't need to recount them all here, I think. I only wrote about the first years of his experience because they are the key as to why the Waffen-SS would name a division after him, and why Belgrade was to be renamed "Prinz-Eugen-Stadt" after the war - Eugene was involved in several battles surrounding the city and successfully managed to conquer it in 1717. In the Nazis' point of view, Eugene had opened up the possibility of German "Lebensraum" by repelling the Ottomans and securing much of the Balkans for Austria, and they wanted to claim his military victories as part of their own tradition.