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Linguistically, "clicks" are a type of sound found in certain African languages, mostly Khoisan and Southern Bantu. The English word is also used for various other sharp, high-pitched noises (like clicking your fingers).

If I wanted a technical term for how click sounds arise in languages (along the lines of "somethingogenesis"), is there an appropriate Greek or Latin root for this?

(Note that I'm interested in clicks as in the type of sound, not as in the action of pressing a button on a mouse.)

There are a couple of Finnish words for a click in the phonetic sense, one of them having a Latin origin.
One of the words we use is avulsiivi, which would correspond to avulsive in English but such a term does not seem to exist in this meaning in English.
The Latin word avulsio refers to the tearing out of a branch of a tree, and the Finnish phonetic term must refer to the corresponding snapping sound.

I am not sure whether other languages use a similar term, but it would be surprising to hear this to be a completely Finnish invention.
This and the other Finnish words are recorded in a well-curated scientific terminology database.

So, perhaps avulsio or something else derived from avellere could work?

OLD defines the noun crepitus as 'A short sharp sound or a succession of such sounds, a creaking, cracking, crashing, clashing, etc.'

This noun and related words are used to cover a fairly wide range of phenomena, such as the rattling of arrows in a quiver, the chattering of teeth, the fall of hail of a roof, the clicking of a bird's bill, the crackling of flames, the snapping of fingers, farts, the creaking of hinges, whip lashes, cymbals, the clapping of hands, and a child's rattle.

This is kind of a "frame challenge" rather than an answer, but I feel like it would be better to just refrain from trying to find a Latin or Greek root to refer to click consonants. When trying to Google for examples of discussion of click consonants in Latin, I came across a passage containing some terrible racism. Unfortunately, there seems to be a history of linguistic racism in the study of Khoisan languages and click consonants: see the following passage from "What click languages can and can't tell us about language origins", by Bonny Sands and Tom Güldemann:

(in The Cradle of Language, edited by Rudolf Botha and Chris Knight, p. 206)

Considering this history, using an uncommon Latin or Greek root to refer to these consonant sounds seems somewhat risky to me, as it could perhaps be understood as indicating that you take a different perspective from mainstream linguistics on the nature of these sounds. Given that "click" seems to be used (with minor adapatations in spelling and pronunciation) in a number of Romance languages to refer to these sounds (e.g. French "clic": Une introduction à la phonétique), I don't think it's necessary to avoid it in Latin. The Vikipaedia article "Linguae Khoisanae" does resort to the formulation "consonantes "clic(k)" ^[en] appellantur" to make it clear what it is talking about.

While it might seem nice to have a single word along the lines of "tonogensis" to refer to the origins of clicks, in English the compound "click genesis" seems to suffice. In an actual Latin text (if you're writing one about linguistics, which I don't think is an extremely common occurrence), I guess some phrase like "origo consonantum clic" might work.