The spelling of "re quite" may seem confusing at first, but it can be explained through its phonetic transcription, /rɪkwʌɪt/. The first syllable, "re", is spelled with an "e" instead of an "a" since it's pronounced as "rih" rather than "rah". The second syllable, "quite", is spelled as such because it's pronounced with a long "i" sound, represented by the letter combination "ui". With this understanding, it's easier to spell and pronounce this word correctly.
The term "re quite" is not a commonly used phrase in the English language, and therefore there is no established etymology for it. It is possible that it is a mistaken or misspelled form of another word or phrase.