late 14c., "a cot, a humble habitation," as of a farm-laborer, from Old French cote "hut, cottage" + Anglo-French suffix -age (according to OED the whole probably denotes "the entire property attached to a cote"). Old French cot is probably from Old Norse kot "hut," cognate of Old English cot, cote "cottage, hut," from Proto-Germanic *kutan (source also of Middle Dutch cot, Dutch kot).
Meaning "small country residence or detached suburban house" (without suggestion of poverty or tenancy) is from 1765. Modern French cottage is a 19c. reborrowing from English. Cottage industry, one that can be done at home, is attested from 1854. Cottage cheese, the U.S. name for a kind of soft, white cheese, is attested from 1831, earliest in reference to Philadelphia:
There was a plate of rye-bread, and a plate of wheat, and a basket of crackers; another plate with half a dozen paltry cakes that looked as if they had been bought under the old Court House; some morsels of dried beef on two little tea-cup plates: and a small glass dish of that preparation of curds, which in vulgar language is called smear-case, but whose nom de guerre is cottage-cheese, at least that was the appellation given it by our hostess. ["Miss Leslie," "Country Lodgings," Godey's "Lady's Book," July 1831]