brat n.

"spoiled and juvenile," 1929, from brat + -y (2).

c. 1300, gyrle "child, young person" (of either sex but most frequently of females), of unknown origin. One guess [OED] leans toward an unrecorded Old English *gyrele, from Proto-Germanic *gurwilon-, diminutive of *gurwjoz (apparently also represented by Low German gære "boy, girl," Norwegian dialectal gorre, Swedish dialectal gurre "small child," though the exact relationship, if any, between all these is obscure), from PIE *ghwrgh-, also found in Greek parthenos "virgin." But this involves some objectionable philology. Liberman (2008) writes:

Girl does not go back to any Old English or Old Germanic form. It is part of a large group of Germanic words whose root begins with a g or k and ends in r. The final consonant in girl is a diminutive suffix. The g-r words denote young animals, children, and all kinds of creatures considered immature, worthless, or past their prime.

Another candidate is Old English gierela "garment" (for possible sense evolution in this theory, compare brat). A former folk-etymology derivation from Latin garrulus "chattering, talkative" is now discarded. Like boy, lass, lad it is of more or less obscure origin. "Probably most of them arose as jocular transferred uses of words that had originally different meaning" [OED]. Specific meaning of "female child" is late 14c. Applied to "any young unmarried woman" since mid-15c. Meaning "sweetheart" is from 1640s. Old girl in reference to a woman of any age is recorded from 1826. Girl next door as a type of unflashy attractiveness is recorded by 1953 (the title of a 20th Century Fox film starring June Haver).

Doris [Day] was a big vocalist even before she hit the movies in 1948. There, as the latest movie colony "girl next door," sunny-faced Doris soon became a leading movie attraction as well as the world's top female recording star. "She's the girl next door, all right," said one Hollywood admirer. "Next door to the bank." [Life magazine, Dec. 22, 1958]

Girl Friday "resourceful young woman assistant" is from 1940, a reference to "Robinson Crusoe." Girl Scout is from 1909. For the usual Old English word, see maiden.