So far as we can tell, his name was Attila. As you've guessed, Huns did not use surnames or family names, and only a non-Hun would call him "the Hun."

Interesting trivia: Attila does not seem to be a Hunnish name but Gothic, like Totila. The Huns were not one tribe but a confederation of tribes including the Goths who did not flee them and the Sarmatian Alans. These tribes tended to adopt Hunnish culture, like cranial deformation and facial cicatrization. But it means that depictions of Attila as a Mongol might be quite a ways off: Goths were Germanic tribes, as in Ostrogoth and Visigoth (tribes that did flee the Huns).

Ingraham, People's Names.