The site stripped the parentheses from the question. It should read: "What is the technical term for the familial relationship you'd have with your half-niece's (daughter of half-brother) son?" Your half-niece is your half-sibling's daughter. So what is the term for your half-niece's daughter, in relation to you? My first best guess would be "half-grandniece", but that doesn't sound right to me, and I guess it with absolutely zero authority.
Hopefully someone can give the correct answer.
Your niece's daughter is your great niece (some would say grand niece).
You are her great aunt or great uncle.
Your first cousin's daughter is your first cousin, once removed.
The technical term for this is getting cold feet.
It's a relationship through marriage so there is no specific term.
Your brother's daughter is your niece and his granddaughter is your great niece.
Your cousin's daughter is called your first cousin, once removed. Although the term "second cousin" is sometimes used for this relationship, it is technically incorrect.
Your daughter and your niece are first cousins. Your daughter and your niece's daughter are first cousins, once removed.
The English language has no collective term that would cover both nieces and nephews, other than the general word "relatives." According to Merriam Webster & Wikipedia; 'niblings' (play on siblings) is a collective term for nephews and nieces.
The English language has no special term for the brother of the woman who married your son.
You daughter's husband's mother is your daughter's mother-in-law. The English language has no term for a relationship between you and your daughter's mother-in-law.
The face is the proper technical term for itself.
unhyperbole
Your brother-in-law's first cousin's daughter is hisfirst cousin, once removed. However, the English language has no term or phrase describing your relationship to that person, with whom you share no ancestry.