It was a long time ago at this point that my daughter asked me why it was called a neighborhood. At the time, I said something like "Uh . . . well . . . um . . . I don't know. Why do you think?" We came up with a nice, totally fabricated story about the houses all being pulled together, like under a hood. But I made a mental note to look it up.
Then forgot. LOL. Until now. ;-)
First, let it be noted that this isn't from hood but from -hood. That hyphen makes all the difference. Hood has been "covering" since the days of Old English, when it was spelled/pronounced "hod." And -hood has meant "state of being" since Old English too, when it was spelled/pronounced "had." So too different OE words that eventually ended up with the same spelling.
So. This made a major light bulb go off. -Hood, as in, motherhood, spinsterhood, etc. This is where "neighborhood" came from, originally meaning "neighborly conduct, friendliness." In the 1620s, it came to be applied to a community of people living close together.
Interestingly, it didn't gain the metaphorical sense (i.e. "Am I close to the answer?" "Eh, you're somewhere in the right neighborhood") until 1857.
So next time my kids ask me this question, I'll have an answer! ;-)