A good place to start with this word (or at least where my Googling had me start) is page 147 in Julio Figueroa G.’s book Vocabulario etimolójico de nombres chilenos (1903), which talks about chilenismos. The very first entry is:
Aconcharse del Araucano Conchu Borra o sedimento
True, that’s a slightly different word (conchu, not gonchu), but it seems too close in meaning and form not to be related.
Aconcharse is in DLE and is derived from yet another slightly differently hispanicised form of the same substrate word, concho:
(Del quechua qonchu, cunchu 'heces', 'posos'.)
- adj. Ec. Del color de las heces de la chicha o de la cerveza. Una mula concha.
- m. Bol., Chile, Col., Cuba, Ec., Pan., Perú y R. Dom. Poso, sedimento, restos de la comida.
The forms mentioned here are in god-knows-what transcription scheme, but I can’t find any exact hits for any of them. Note also that the language of origin is different: Quechua as opposed to Araucano (which we nowadays call Mapuche). Quechua and Mapuche are not related (Mapuche is a language isolate). Either one of them borrowed the word from the other; or either Figueroa or DLE gives the wrong source language.
Which of the two is the correct interpretation, I do not know. But it seems that qunchu is the Quechua word for yeast; specifically, aqha qunchu is saccharomyces cerevisiae, a particular type of yeast that has been used in brewing and wine-making since ancient times and is commonly known as brewer’s yeast in English. When making wine and beer, the yeast that hastens the fermentation process also creates a layer of sediment at the bottom of the vat, known in English as lees (I don’t think Spanish has a specific word here—just posos/heces de vino/cerveza), or trubs in beer-making.
According to AULEX (an online Quechua-Spanish dictionary), qunchu also means ‘dregs’ in Quechua itself, like in Spanish.
I’m not familiar with any etymological dictionaries of Quechua, but my guess would be that the original meaning is ‘(brewer’s) yeast’, which was then extended within Quechua to also refer to the lees/trubs at the bottom of the vat where alcohol was fermented. It was this latter meaning which was borrowed from Quechua into Spanish as concho (and perhaps conchu and/or cunchu as well). Or possibly from Quechua into Mapuche, and then from Mapuche into Spanish. Or possibly both.
The Spanish variant that begins with /ɡ/ instead of /k/ (i.e. ⟨gonchu⟩ instead of ⟨concho/conchu/cunchu⟩) is a bit of a mystery to me, though. Neither Quechua nor Mapuche possesses voiced consonants, so whichever of the two is the immediate source of the Spanish word, the development /k/ → /ɡ/ happened within Spanish itself. I am not aware of any pattern or reason why this voicing should take place, but it seems that it did.