In Colombia, there is the Cócora Valley.
A branch of the Cepeda family is better-known-as the familian Kókora.
The word is Spanish, cócora, and it means "an irritating, annoying or fussy person."
Chamorros love to tease, and pick on a particular trait, physical or personal, of another person. It's possible an ancestor in this family had this trait, or was just humorously picked on and tagged with this label. It certainly does not describe the whole clan, any more than everyone smells of onions in the Seboyas family or everyone is blind in the Båtchet family.
We have to allow also for other possible origins of this nickname.
Cócora is also a slang word in some parts of Mexico. Also in Portugal and Brazil.
There is also a valley in Colombia named Cócora.
Is it possible that Cócora is Chamorro, and not borrowed from Spanish or Portuguese? It's possible, but unlikely in my opinion since Chamorro seems to lack the R sound (at least our language usually changes the R to an L). Our language usually stresses the penultimate (next-to-the-last) syllable, not the first as in Cócora. And, since there are numerous examples of Cócora appearing in other languages, there's a better chance, I think, that the word is from these other languages.