"Booyah Base" is a play on the word "bouillabaisse," a type of fish stew from southern France. Its name was based on the Provençal Occitan language word "bolhabaissa", which was derived from the verbs "bolhir" (to boil) and "abaissar" (to simmer or reduce heat).
It is also a pun on Shibuya(Japanese: 渋谷区), a city within Tokyo, Japan which it (especially its shopping district) is influenced by.