Hungarian, Turkish or one of the Northern Turkic dialects (I prefer Kumyk) would be accurate for Huns. I will explain why.
First of all most of the posters focus on Attila which only explains his language in particular. Attila was well educated leader who visited Rome and maintained strong relationship with Germanic tribes. His knowledge of languages at that time period of Europe was over average, it can even rival Roman elites.
However Huns were confederation of nomadic tribes dominated by Altaic-Turkic tribes. After the collapse of Hun Empire in the Asia (Asian Hun Empire:
http://bit.ly/w8UuDM), tribes migrated to the steppes in north of Black Sea (usual route for Turkic tribes since Scythians) where they met with their far relative Finno-Ugric tribes (Ural branch of Ural-Altaic language group) dominated by Magyars. It must be that period of time when Huns learned to build wooden constructions in forested areas and integrated with Finno-Ugric culture.
This is why when they came to Alföld plains (another route for Turkic tribes), they had a language similar in today's Hungarian and Turkish language. But in either case the language was certainly Ural-Altaic. I assume first period of Europen Hunnic domination language was more like Turkic but in time, perhaps before collapse of the Empire, language was more similar to Hungarian (in two hundred years approximately). It would be historically accurate to choose Northern Turkish dialects (Kazakh, Chuvash, Kumyk) or Hungarian.
I rely on facts and some assumptions summarized above, to support my argument. I can write further detail but I wanted it to be short.
For other language offers for Huns, I can only say they are either too much biased or incredibly ignorant.
For Gothic (so called Lingua Franca). It is known that Huns defeated Ostrogoth when they were attacked by them and that attempt to destroy Huns triggered a series of events which resulted in Fall of Rome. One hundred years of Goth expansion conquered in 3 years. Goths were first to met devastating power of Hun army and they were first to migrate far reaches of Roman empire. It was never Lingua Franca for any of tribes except for some of Germanic tribes, certainly not for Huns. Plus if there was ever international language at that period it was Latin which most of military and administrative elites of different ethnic groups were eager to learn as Huns did. Attila's father, Mundzuk (Boncuk in Turkish, means "pearl") and his grandather Uldiz (Yıldız in Turkish, means "star") might have familiar with Goths but I doubt they knew Gothic language.
For Mongolian. Contrary to general misunderstanding Mongolian was not so much different until son of Genghis Khan, Kublai Khan established his Khanate in Far East of Asia. That geographical separation led them culturally differentiate in 600 years of process. Other parts of Mongolian Empire (Golden Horde, Caghatai Khanate, Il Kahante) quickly fell under Turkish domination as their majority in army and population were Turkish but Kublai Khan's Khanate adopted Chinese and Uighur (another Turkic tribe, Mongolian writing is copy of Uighur) culture. So it would be ethnically, geographically and chronologically wrong to choose Mongolian.
It is very hard to choose language for once powerful but destroyed civilization. It is even harder when they speack Ural Altaic language which spreads across Europe and Asia.
In final analysis, no matter how much they are distant to modern cultures their culture is related to Ural-Altaic sphere which includes both Finno-Ugric and Turkic culture. I know it is long to read but I felt I should give some information about one of the mightiest Empire of the history. It would be unfair to forget their impact on world.
I would gladly listen if there is any misinformation, I humbly wrote only what I know from my personal interest on history in general, and Turkic history in particular. Also I apologize for my English for it is not my mother tongue, I would be sorry if there is any misunderstanding because of it.
Note: The word "Hungar" doesn't derive from Hun. It comes from word "On Ogur" which means "Ten Ogurs". Ogur, Oguz (or Oghuz), Ugor, Uighur are different versions of same word which can be observed in the most of the Ural-Altaic tribe names. Ten Ogurs/Ugors might have been dominant Boy (tribe) in Magyar Uruk (tribe federation) to persist through history.
. But yes, I agree with you, It would be more accurate to choose Altaic language for Attila.
