According to Stephen Esposito's notes in his translation of Euripedes Bacchae, the word is potentially Lydian in origin. There are a couple competing theories for the origin of the Greek bakkhos. One argues it's related to bacca, a Latin word for berries or fruits of trees, or could potentially have been adopted from an unidentified Asian language.
Important to note that the cult figure of Dionysus went by several different names. In the prologue of Bacchae, there's a nod to both possible origins of his name:
"After leaving the gold-rich fields of the Lydians... I moved on to Persia's sun-parched plateaux... and all of Asia Minor... I first came to [Thebes] only after I had roused to dancing all those Asian lands and established my rites there so that I might be seen by mortals as a god."
In Bacchae, the god is simultaneously referred to as Bromios (the roaring god), Bacchus, and Dionysus (a play on words: Dios from the possessive case of Zeus's name and emphasizing his descent from the god. Nysus is less clear and folk etymologists believe it derives from local worship of Zeus of Mt. Nysa, Mt. Nysa being the mythic location where Dionysus was raised).
Throughout the play, we see a multifaceted god whose domain is the bull, earthquakes, the common man, frenzy, the grape vine, ivy, liberation, lightning, liquid nature, self-transformation, trees, theater, and -- naturally -- wine. This is common given the syncretic nature of worship of many ancient gods that were conglomerated into centralized figures.
Source: Euripides, & Esposito, S. J. (1998). Euripedes Bacchae. Focus Pub./R Pullins Co.
Just adding...
I don't know if there's some consesus on the word-origin of Bacchus [Βάκχος].
Many dictionaries are mentioning it as a word of unknown origin, while it has been noted the Lydian relation or even a greek one with the older word-exclamation βαβαί [check eg Frisk, 1960, p. 212 in https://archive.org/details/hjalmar/page/n241/mode/2up?view=theater , where: "Fremdwort unbekannter Herkunft"].
Or even another greek origin from a word-root for say-shout-cry [eg. Bailly, 1935, p. 344 in https://archive.org/details/BaillyDictionnaireGrecFrancais/page/n343/mode/2up ]. Though it isn't directly mentioned, this latter greek origin, had been suggested already in the 12th c. CE by Eustathios of Thessalonica in his Iliad Commentary, with similar word-roots and related to Iacchus [Ίακχος] [check in greek, 6. v. 132-137 in http://www.poesialatina.it/_ns/Greek/testi/Eustathius/Commentarii_ad_Homeri_Iliadem02.html ]
Regarding the Lydian relation I didn't track a certainty if it's a loan from or a loan to. The main older argument seems coming from a bilingual Lydian-Greek inscription of the Persian age [dated possibly in 350 BCE] and related to a Lydian word 'baki' for Dionysus [info via Frisk that more possibly is in https://sardisexpedition.org/en/artifacts/r2-274 ]. I don't know of newer findings on this Lydian relation but I've read somewhere for a similar Lydian word 'bakillis' that stands for the priest of Dionysus.
Regarding the ancient Greek literature, though the word Dionysus appears in Homer of the 8th c BCE, the Liddell-Scott dictionary is giving as first appearance of the word Bacchus [Βάκχος], the tragedy by Sophocles, Oedipus Tyrannus [v. 211] of 430 BCE ca. The word there is a noun - name. However the root can be tracked earlier in the 5th c. BCE as an adjective-epithet attached to Dionysus and as a verb [: bacchios Dionysus=Βάκχειος Διόνυσος, baccheuo=βακχεύω]. Example is Herodotus [4.79 in http://www.perseus.tufts.edu/hopper/text?doc=urn:cts:greekLit:tlg0016.tlg001.perseus-grc1:4.79 with a 1920 translation]. Also in the Homeric Hymn 19 to Pan, again as an adjective attached to Dionysus. The latter hymn's dating is uncertain; more possibly of the early 5th c. BCE.
I don't know if this sequence of the appearances in the ancient greek language [i.e. firstly as an adjective and verb in Herodotus, then as a noun-name in Sophocles] truly reflects reality and could indicate if the word root was firstly just an attribute attached to Dionysus, and not a separated name-god that was imported in Greece.