According to The History & design of the Australian House
The name of the town was derived from the Aboriginal Araba tribal words kupa piti meaning 'boy's waterhole', and not 'white man in a hole' or 'white fellow burrow', as is often mistakenly recorded
and the 1969 book Opal Men says:
The translation usually given is "whitefellow burrow" but this is incorrect. The South Australian Museum's dictionary of the Arabana dialect gives it as "boy's waterhole" — the one used by lads who had not been through the ceremony which initiated them into manhood.
However, the most authoritative article on this topic is The etymology of Coober Pedy, South Australia Aboriginal History Vol. 34 (2010), pp. 217-233.
This article says the name Coober Pedy was selected by vote of 16 non-indigenous people after asking indigenous people "what do you call these diggings?".
The article concludes that the name comes from "kupa" meaning "whitefella" and "piti" meaning "quarry". The word "kupa" can mean "boy who has not been initiated by circumcision ceremony", but this term was also used for white men, since they were not initiated by such a ceremony.