Origin of "quarters" in the sense of living area

Solution 1:

Etymonline has this to say on quarters:

"military dwelling place," 1590s, from quarter (n.) in sense of "portion of a town." The military sense is in quartermaster (mid-15c.) and might be behind the phrase give (no) quarter (1610s), on the notion of "to provide a prisoner with shelter." The verb quarter "to put up soldiers" is recorded from 1590s.

Quartermaster appears to be from something unrelated to fourths:

mid-15c., from Fr. quartier-maître or Du. kwartier-meester; originally a ship’s officer whose duties included stowing of the hold; later (c.1600) an officer in charge of quarters and rations for troops.

Although I am weak at French, it appears that quartier means area or district. Note that the dates for quartermaster are before quarter: mid-15c versus 1590s. This implies that quartermaster transferred first and then people began using quarter due to the use of quartermaster.

Transferring the meaning of quarters into a non-military dwelling place seems reasonable and, therefore, the short answer to your question is that quarters appears to have come from France by way of quartermaster.

Edit: My hunch on why quarter came from quartermaster (as opposed to Etymonline just telling us it did) is the inclusion of master in the term led people to assume that they were masters of something; it was logical to call that something a quarter. The idea of quartering up soldiers would mean to provide for them; this would be the quartermaster's responsibility. Lodging was most likely included in that and the perception of "master of lodging" makes sense.

Solution 2:

Yeah, the process by which quarter goes from meaning "divide into four parts" to a "portion or a part" has analogues. It's a sort of semantic slide or shift that I love discovering. In a particular Spanish-language text I was reading, a speaker commands his listener, Adocénate! It literally means, "Dozen-ize yourself," which led me to think, "Well, that's odd." But further reflection led me to see that it was part of an exhortation to the listener to multiply his efforts, his presence, and so on. Quarter has apparently undergone something of the same process.

So quarter goes from meaning to divide into four portions, its plain Latin origin, to meaning either one of those four portions, or to divide into portions generally. From the idea of those four portions, it comes to mean any portion. The military portion is the military quarters, just like the French part of town is the French Quarter, as in New Orleans. A smaller and smaller area will suffice as one's personal quarters, until quarters comes to mean lodging in general, though with a military connotation. It might not only be lodging, or lodging in particular, but a station generally - including a battle station. In the US Navy, they call it general quarters - where you are to position yourself for readiness in battle. The British Navy, our own antecedent, spoke of beating to quarters because a drum was used to transmit the command to go to quarters. From there, it is a short shift to the chief's quarters, which naturally will include study and meeting spaces, a headquarters, if you will.

Great fun question, isn't it? I hope your son got a cookie out of the deal :)

Solution 3:

The NOAD reports that quarters means "rooms or lodgings, especially those allocated to servicemen or to staff in domestic service." The Online Etymology Dictionary reports that quarters, used to mean "the portion of a town identified by the class or race of people who live there," is first attested in 1520s.

The word was then extended to also mean "the portion of a house identified by the class or the race of people living there."