What do “Háni" and “Sons éso tse-ná” mean in Brave New World? I know they sound unreal, but I was curious if there is an actual meaning behind these two phrases.
Both phrases are borrowed from Frank Hamilton Cushing's Zuñi Folk Tales (G. P. Putnam's Sons, 1901).
Below is a passage that uses the word "háni" (in Átahsaia, the cannibal demon):
One bright morning in summer-time, the elder sister called to the younger, "Háni!"
"What sayest thou?" said the háni.
Based on this, háni would seem to mean "younger sister".
Below is a passage that uses the phrase "Sons éso tse-ná" (in the introduction to "How the twins of war and chance, Áhaiyúta and Mátsailéma, fared with the unborn-made men of the underworld"):
"Did you hever hear tell of the people who could not digest, (...)?"
"Nay, never; not even from my own grandfathers," said I. "Sons éso to your story; short be it or long". 
"Sons éso tse-ná! ("Cool your 'sons éso!' and wait till I begin)
 A footnote explains,
The invariable formula for beginning a folk tale, by the raconteur: "Són ah-tchi!" ("Let us take up")--té-la-p'-ne, or "a folk tale,", being understood. To this the auditors or listeners respond: "É-so!" ("Yea, verily.") Again, by the raconteur: Sons i-nó-o-to-na! Tem," etc. ("Let us (tell of) the times of creation! When," etc) Again by the listeners "Sons éso! Te-ä-tú!" ("Yea, let us, verily! Be it so.")
Most readers of Brave New World would be unfamiliar with the Zuñi language and might therefore conclude that háni represents an insult or a curse and that Sons éso tse-ná is another expression of his anger. This would be based on clues such as "with what derisive ferocity!" and "it was lucky that Bernard didn't understand Zuñi". From this point of view, John's reversal to Zuñi may be seen as a symptom of regression.
Of course, there is a form of irony here that readers unfamiliar with Zuñi will not notice: using a kinship term meaning "younger sister" in a society that has abolished the family can be seen as a scornful criticism on that type of society.