To clarify, I'm asking for historical reasons why we can't pinpoint the identity of the Huns.
They are not certainly not unique in being a group we only know about through historical records written by outsiders. There are dozens of groups like that whose identity we can surmise.
They are also not the only group whose language only survives through a handful of words, mostly names of people and places. The only real records we have of the Cimmerian language are proper names but we still identify them as Indo-European, even if we can't classify them any further. We at least have a pretty widely accepted theory about the Aquitani being the ancestors of the Basque people, even though we only know their language through names on funerary slabs. The Scythian language is almost completely unattested but we still identify them not just as Indo-European, but specifically as speakers of an Eastern Iranian language.
The Huns also occupied a pretty large territory, had a lot of contact with the Romans and other literate groups, and there are records of their presence in the Eastern Empire for almost 400 years. We know that they signed treaties and made diplomatic relations with Romans, we have artifacts of their material culture, and we have a lot of knowledge about their way of life, and possibly even their religious beliefs.
So why are historians not able to classify the Huns, or determine their relationship to other groups? There are various proposals that they're Indo-European (maybe Germanic), Turkic, Mongolic, Yeniseian, Tungusic, somehow related to the Xiongnu, or an isolate. Why can't we confirm or discredit these theories? What's keeping us in the dark?
First, we have to distinguish Huns as a people, from the Hunnic confederation they dominated in the early Vth century Barbaricum from Rhine to Black Sea. Rather than an empire or kingdom, this Hunnic hegemony took the form of a vast coalition while Huns themselves mostly settled in Danubian regions, critically in the modern Hungarian plain during Attila's rule. The collapse of this hegemony will leave no real Hunnic legacy in most of these people because there were not much in the first place.
Paradoxally, looking at the core of the Hunnic "empire" doesn't really helps understanding what were the speficic cultural features of the Huns. We have to take a look at their history, as they entered into european Barbaricum : the identification of Huns with Xiongu isn't necessarily wrong (even if this association often led too easily to paint Huns as pseudo-Mongols) but the geographical and chronological distance is quite a problem.
Linguistic evidence, from the limited pool of attested Hunnic words : while some could be attributed a turkic or mongolic origin, ugrian, sarmatic or even germanic elements could be argued. This is a testimony of the important cultural mixity of Barbarian peoples before their entry in Romania and especially in steppes where material archeological cultures can't really be attributed to a specific cultural group in spite of many tentatives in the XXth.
The Xiongu hypothesis is mostly based on historical accounts and historiographical interpretations : evidence of destruction of fortifications in Turkestan in the IIIrd century could be tied up with Xiongnu which were accounted for having taken over part of these regions, and whom a king was named "Huni". Medieval Ugrian Hunni living east of Volga and Late Ancient Chionites living in Turkestan could be a legacy of a Xiongu-Hunnic presence in these region, especially as an central asian origin of some material evidence in finds associated with European Huns is attested.
But these evidences aren't particularily Turkic or Mongolic, but rather Sarmatic and Ugrians : while Huns could have inherited some Turkic or Mongoloid origin is quite possible if not likely, but as they appeared as a distinct people in Eurasian steppes, they simply might have been right from the start a metis people with a lot of various origin, greatly influenced by Sarmatians and Ugrian features, possibly proto-Slavic too.
Why don't know more about this? Well, they were newcomers in Europe, and Romans never had a clear idea what were people living beyond Germania, not just due to distance, but as well a certain conception of Barbarians as being without history : such as they were, they remained (for instance, Vandals were considered Celtic until the IIIrd century by some Roman scholars). Central Asia was definitely too far of Roman focus; and when they entered in Europe and whtin Roman sphere, they were identified and defined "politically" and genealogically rather than according their language or "ethnics" as we now understand the word.
The difficulty might furthermore come from the reality of cultural mix-up of steppe confederations, even more marked than with other Barbarians and happening in the long term.
And, eventually, the quick collapse of the Hunnic hegemony, and the disappearance of Hunnic peoples in the VIth century (while Aquitains or Sarmatians were known over a relatively long period) most probably represented another obstacle.
Sources
- Les Huns, le grand empire Barbare d'Europe, IVème-Vème siècle; Istvan Bona
- Des Goths aux Huns, les bords de la Mer Noire au Bas-Empire et à l'époque des Grandes Migrations