Ultimately from Sanskrit बुद्ध (buddha, “awakened, enlightened; the Buddha”).
The first two syllables must certainly reflect Middle Chinese 浮屠 (bɨu duo, “Buddha”) or a similar Chinese form for "Buddha". Thomas Pellard connects the final syllable to 加 (MC kae), a common element in the noble titles of the Buyeo and Goguryeo kingdoms of ancient Korea, and to 皆 (MC keaj), appearing in pseudo-Goguryeo toponyms with the apparent meaning of "king". Pellard thus hypothesizes that the Korean word originally meant "Lord Buddha" or "King Buddha".
Compare the identical semantics of Old Uyghur 𐽼𐽳𐽾𐽲𐽰𐽺 (burhan, “Buddha”), composed of 𐽼𐽳𐽾 (bur, “Buddha”) and 𐽲𐽰𐽺 (han, kan, “ruler, Khan”), with the first element in the compound also being from Chinese origin.
*Pwutukye or *Pwutokye
The reconstruction is supported primarily by the Manchu and Japanese borrowings; comparative reconstruction within Koreanic only goes back to Middle Korean 부톄〯 (pwùthyěy). However, the exact consonantal match between Jurchenic and Japanese cannot be explained historically without positing a Koreanic source.
To explain the differences in vowel quality between the three languages, Pellard notes that Old Korean very possibly lacked vowel harmony (which is in fact never written in Old Korean orthography), and that Middle Korean <ye> /jə/ may derive from Old Korean /e/. Therefore, the word reconstructed in Yale romanization as <Pwutukye> may actually have had the phonetic value */putəke/, which matches Old Japanese <poto2ke2> */pətəkəɨ/ quite well.
While universally agreed to be a Koreanic borrowing, the Manchu form is in fact more problematic than Pellard supposes. Pellard believes that Jurchen would have borrowed */e/ as /i/ because Manchu <e> has the phonetic value /ə/, but this is really the result of a Manchu sound shift from */e/ > /ə/; thirteenth-century Jin Jurchen did in fact have */e/. In all likelihood, the Koreanic donor to Jurchenic, probably the speech of Goguryeo, had a different vocalism from the Baekje donor to Japan or the Old Korean ancestor of Middle Korean.
The Japanese and Korean etymologies have traditional Chinese-based etymologies, but both are less likely than the Koreanic explanation.