I don't know about inspirations and I'm not sure how you would set about researching it, but I've been keeping note of names and any stated translations so if there is a genuine language that it's based on, this would be a good start for cross-checking.
NAMES
It is unlikely that the names are gendered and there are not enough to be worth separating the list. Elene'shpya, Kaahe and Shunye are male characters; the rest are female.
Apyaahi
Cahciua
Ecaate
Elene'shpya
Elenyate
I'toca
Iyaate
Kuiyki
Naahe
Nahueni
Oshce
Shaloaci ("Shalo")
Shashepya
Shunye
Taapu
Toashiua
Uaauashe
Uah'shepya
Uapyota
Ui'she
Yyopya
Yyopyicana
Yyupye
Additionally, there are a small number of names in the levequests that are one or another of the Xak Turali races but not indicated which -- these are Uapyone, Pananyihe, Shoape, Yimepi and Yoani. I would guess that Uapyone has to be a Shetona name, Panayihe might be Tonawawta and the other three may be Hhetsarro.
CREATURES
Yeheheceyaa (Pteranodon)
Lunyuhiyshahe (Triceratops)
Lunyucaua'pya (Tyrannosaurus)
Yehehetoaua'pyo (Shaaloani elite mark, pteranodon) = words for "blue", "wings" and "extinction" [*1]
Keheniheyamewi (Shaaloani elite mark, chameleon) = "mouth that breathes poison" [*1]
Heshuala (Heritage Found elite mark) = "child of thunder" [*2]
Yyenisheyni Bat (found in caves in East Yyasulani)
Cauahealoa (werewolves/wargs in East Yyasulani)
Cauahepya (wolvesgnolls in East Yyasulani)
Elene'shpya = mallard, but probably has a more literal translation. Also tells us that animal names can be used as person names.
Shshuye (jackrabbit minion) = "leaping sun"
Lunipyati (jackrabbit tural vidraal)
LOCATIONS
Pyaayehe'pya = "hill of many wings", nesting ground of the Yeheheceyaa (Pteranodon)
Yyasulani = "where the clouds dry" [*3]
Tesh'pyani = "where the stones howl"
Eshceyaani Wilds (region of Shaaloani)
Shaaloani itself is likely to be a Shetona name, given the recurring "ni" ending for locations.
Niikwerepi River -- uncertain if this is a Shetona name: it is mentioned as important to them in the BTN/MIN questline, but it is located in the region that seems to have mostly Hhetsarro names.
Sheshenewezi Springs -- also uncertain
MISC
Aloauanyeyci -- this is written on the levequest icon showing a Shetona, equivalent to Eorzean "piety" leves.
Uktena -- Shaaloani elite mark, a snake. Uncertain which local language it is named in.
SOURCES
1. Hunt descriptions from Elenyate in Hhusatahwi
2. Hunt descriptions from Loanyine in the Outskirts
3. Quest: "Yyupye's Dirt"
4. Quest: "What She Left Behind" - https://www.garlandtools.org/db/#quest/70698