The prefix "не" is used to negate or reverse the meaning of the word:
вежливый (polite) → невежливый (impolite)
слабо (weakly) → неслабо (strongly)
Following this logic, "нечто" should be interpreted as "nothing," but the actual meaning of "нечто" is "something."
The word "нехай" means "may it happen that ...," so it is unclear as to what is negated by "не" in this word.
My question is this: What does "не" negate in "нечто" and "нехай," or how can you explain the presence "не" in these words?
P.S. I found a question about "нечто," but my question is different because it is not about the meaning of the word, but about the composition of the word, specifically about the logical role of "не".