avant-hier = literally "The day before yesterday."
I'm trying to figure out why this isn't as cool to me as the Japanese words 
ototoi and 
asatte. At first, I thought it was because the words weren't related to the words 
yesterday and 
tomorrow at all, but I looked at the kanji and it turns out they are.
?? = kinou = yesterday

 = ototoi = day before yesterday (lit: one-previous-day)
?? = ashita = tomorrow

 = asatte = day after tomorrow (lit: bright-after-day)
Those aren't cool and original at all.... only their readings. My guess is that native Japanese already had pronunciations like 
ototoi and then when the Chinese introduced writing, they just assigned some kanji to go with it, rather than give it a new name based on the Chinese pronunciation of the kanji.