At least from a synchronic perspective, Hannaito seems to mean something like "Fighterish" or "Warriorese". This is how the name of the language is most likely to be interpreted by its speakers, who likewise call themselves Hannaichë "Fighter(s), Warrior(s)". However, the Hannaichë themselves tend to find the perceived connection between their autoethnonym and the verb hannai "to fight; to battle" a bit strange, although the majority of them rarely give it much thought in their daily lives. While no culture is truly monolithic, of course, the Hannaichë still share a common ethnic identity, a general idea of who they are, collectively, and how they'd like to perceive themselves as a group. The confusion regarding the apparent etymology of their endonym stems from the fact that, although they are certainly not strict pacifists, the Hannaichë prefer to see themselves as belonging to a nation of artists and musicians, not warriors, above all else. In other words, this is a culture that views writing a well-received song about a military victory as an even more glorious and prestigious accomplishment than actually participating in the battle itself. Additionally, there is no clear evidence from either their written or oral history, folklore, or mythology that Hannaichë ethnic identity has ever been primarily focused on military prowess in the past. As a result, the assumption that the name of the language and the speakers' name for themselves are transparently derived from the verb hannai "to fight; to battle" has been called into question, and so the etymology of Hannaito and Hannaichë is generally considered a matter of debate.
It is possible, of course, that things truly are as they seem - in other words, that Hannaito and Hannaichë are actually derived rather straightforwardly from hannai "to fight; to battle". Some are content to leave it at that, preferring to believe that the the simplest, most obvious solution is, in fact, correct and that the cultural attitudes described above are irrelevant and should not prevent this etymology from being accepted. Proponents of this theory argue that the Hannaichë do not need to have ever valued war above all else in order to have begun calling themselves the "Fighter(s), Warrior(s)". Meanwhile, many others accept a connection between the names Hannaito and Hannaichë and the verb hannai "to fight; to battle" without dismissing the need to address the associated cultural "issues". Perhaps the most widely accepted theory is that Senses 11-17 of hannai in the modern language are original, and that the remaining senses of the word, despite being used more frequently and generally thought of as more "basic" or primary than Senses 11-17 today, only developed later, after speakers of the language had already begun calling themselves Hannaichë. According to this theory, the Hannaichë are thus the "Fighters" in the sense of "those who struggle yet persist against the odds" rather than "those who wage war", which is much more in line with the stories they tell about themselves and their origins as a people. The fact that Proto-Hannaito *haŋkay is translated as "to struggle, to strive, to endure, to toil, to try" above reflects the popularity of this theory.
However, it seems possible, if not likely, that *haŋkay was already used to mean either "to struggle, to endure" or "to battle, to brawl" in Proto-Hannaito, with neither sense being a later development. In the earliest written records of Hannaito, hannai was seemingly used to describe both physically engaging in combat and spiritually persevering through hard times, although there is not enough evidence to determine whether or not one meaning was more common or more "basic" than the other. Cognates of hannai in other "Hannaitoan"(?) languages do not provide much help in this regard, as there seems to be a fairly even split between languages in which the reflex of *haŋkay is primarily used to mean "to fight, to battle" and those in which it is primarily used to mean "to struggle, to strive, to endure, to toil, to try". Of course, there are also a few members of the family which have no known reflex of *haŋkay at all or in which the reflex of *haŋkay has clearly undergone a fair amount of semantic drift. In any case, it is noteworthy that no other "Hannaitoan"(?)-speaking group seems to have derived its autoethnonym from a reflex of *haŋkay or any other word connected to "fighting" or "war".
Another theory is that Hannai- and hannai "to fight; to battle" are etymologically unrelated, but have became homophonous due to regular sound change and are no longer seen as distinct due to the fact that, in the modern language, Hannai- only appears as the root of words like Hannaito and Hannaichë, while the verb hannai is quite common on its own. In addition to *haŋkay, there are indeed a fair number of other hypothetical Proto-Hannaito word forms which would be expected to yield hannai in modern Hannaito - including *kaqnay, *gaŋhayi, and *hagmaqi, just to name a few. However, there is no clear evidence that reflexes of any of these hypothetical words, other than *haŋkay, have survived in modern Hannaito. Even if they had, none of these words would likely make more sense semantically than *haŋkay as the ancestor of the ethnonym Hannai-, given the meanings of their reflexes in other "Hannaitoan"(?) languages. For instance, *kaqnay, *gaŋhayi, and *hagmaqi can be reconstructed as having originally meant "thorn", "squirrel", and "to choke", respectively.
It is alternatively possible that the element Hannai- was originally something like Honnai-, Handai-, or Hannae-, but was later altered to match the more common hannai "to fight; to battle" via folk etymology after falling out of use outside of words like Hannaito and Hannaichë. However, there is no direct evidence of this, as hypothetical forms like Hannaeto or Honnaichë are completely unattested. Additionally, words in other "Hannaitoan"(?) languages that could be cognates of hypothetical Hannaito words like Honnai-, Handai-, or Hannae- are either nonexistent or hard to imagine as ethnonyms on semantic grounds. For example, reflexes of Proto-Hannaito *gannoqê, which would be expected to yield hannae in modern Hannaito, typically mean something like "grub, larva", and *hetnayi, which would be expected to yield handai in modern Hannaito, cannot be reconstructed for Proto-Hannaito, despite being a theoretically acceptable word form, based on available evidence from its modern descendants.
Finally, it has been suggested that names like Hannaito and Hannaichë may be, in some way, the result of language contact between speakers of an early form of Hannaito and speakers of a non-"Hannaitoan"(?) language. If so, this development likely took place after "Old Hannaito" had already split off from the rest of the "Hannaitoan"(?) languages, due to the fact that the element Hannai-, as an ethnonym potentially distinct from the reflex of *haŋkay, seems to be unique to Hannaito, but before "Old Hannaito" was first written down, since forms ancestral to Hannaito and Hannaichë, for instance, are attested fairly early on. During this period, "Old Hannaito" was in contact with languages belonging to two other families, Leran and Shoic (very tentative names). More specifically, speakers of "Old Hannaito" came into contact with speakers of Proto-Shoic, which is unattested, and "Western Leran", which is attested.
Despite the large number of Hannaito words of Shoic origin, Hannai- is unlikely to have originated as a direct borrowing from Proto-Shoic for phonological reasons. The source of Hannai-, if it had been borrowed from Proto-Shoic, would have to have been something like *ɦannãj, but such a word could not have existed in Proto-Shoic due to its violation of nasal harmony. The closest "legal" word form would have been either *ɦaddaj or *ŋãnnãɲ, but those likely would have been borrowed into Hannaito as Hattai- and Nannan-, respectively. Meanwhile, written records of Western Leran from this period indicate that it did not have any constraints prohibiting the existence of words that could have been borrowed into Hannaito as Hannai-. However, given their attested meanings at the time, the meanings of their modern descendants/reflexes, and the meanings of their cognates in other Leran languages spoken farther afield, none of these words would seem to make much sense, semantically, as the source of Hannai-. Also, the majority of Hannaito words of Western Leran origin appear to have been borrowed after this period.
Furthermore, although it is not completely unthinkable for a group's autoethnonym to contain an element originally of foreign origin, that seems unlikely to be the case here. Assuming that Hannai- is of non-native origin and thus unrelated to hannai "to fight; to battle", there is no evidence that it has ever been used on its own or outside of words referring to Hannaito and the Hannaichë. Also, as mentioned above, forms containing Hannai- are attested fairly early on in Hannaito's written history, not long after the period of time during which Hannai- would have to have been borrowed from an unrelated language. In light of all of this, if Hannai- indeed originated as a direct borrowing from Western Leran or Proto-Shoic, it must have been adopted as an ethnonym rather quickly, seemingly without being used in any other context. It seems very unlikely that the Hannaichë borrowed a word or part of a word seemingly for the sole purpose of almost immediately beginning to use it as their name for themselves.
Due to the unlikelihood of Hannai- being a direct borrowing from another language, the majority of those who support the idea that names like Hannaito and Hannaichë are, in some way, the result of language contact believe that that these current endonyms originated as calques of Shoic exonyms. According to this theory, the element Hannai- is etymologically identical to the verb hannai "to fight; to battle", and Hannaichë was indeed originally intended to be interpreted as "Fighter(s), Warrior(s)". After "Old Hannaito" split off from the rest of the "Hannaitoan"(?) languages, but before it was first written down, the language's speakers migrated from the family's mainland Urheimat and onto a nearby archipelago originally inhabited by the speakers of Proto-Shoic. Based on archaeological evidence and oral history, this "migration" was not an entirely peaceful process, but the majority of the islands' original population seems to have survived. In modern times, there are prominent references in pan-Shoic folklore to some sort of "invasion", "conquest", or "war" in the nebulous, mythic past. While most speakers of modern Shoic languages do not associate these stories, either explicitly or implicitly, with the Hannaichë, it seems highly likely that they were inspired by the historical "Hannaitoan"(?) migration onto the archipelago where the speakers of Proto-Shoic lived. There are no records of what Proto-Shoic speakers called the Hannaichë during this early period, but proponents of this theory believe that they referred to their new neighbors as the "Invader(s), Conqueror(s), Warrior(s)", an epithet that was calqued into Hannaito as Hannaichë "Fighter(s), Warrior(s)", perhaps as part of an attempt by the Hannaichë to make themselves seem more intimidating as they began to exert control over new territory. According to this theory, as their relationships with and attitudes towards the Hannaichë improved over time, speakers of the Shoic languages stopped calling them "Invader(s), Conqueror(s), Warrior(s)", while the Hannaichë continued to call themselves "Fighter(s), Warrior(s)" and have now long since forgotten why they started doing so.