late Old English scoru "twenty," from Old Norse skor "mark, notch, incision; a rift in rock," also, in Icelandic, "twenty," from Proto-Germanic *skur-, from PIE root *sker- (1) "to cut."
The notion probably is of counting large numbers (of a passing flock of sheep, etc.) by making a notch in a stick for each 20. The prehistoric sense of the Germanic word, then, likely was "straight mark like a scratch, line drawn by a sharp instrument." That way of counting, called vigesimalism, is widespread and also exists in France and left its trace in the language: In Old French, "twenty" (vint) or a multiple of it could be used as a base, as in vint et doze ("32"), dous vinz et diz ("50"). Vigesimalism was or is a feature of Welsh, Irish, Gaelic and Breton (as well as non-IE Basque), and it is speculated that the English and the French learned it from the Celts. Compare tally (n.).
By early 13c. it is attested in the sense of "a financial record" (perhaps one kept by tallies), and it is attested from early 14c. as "reckoning, total amount." The specific sense of "a reckoning or account kept by means of tallies" is clearly attested by c. 1400, especially (1590s) "mark made (by chalk, on a taproom door, etc.) to keep count of a customer's drinks."
This was extended by c. 1600 to "amount due, one's debt," and by 1670s to "mark made for purpose of recording a point in a game or match," and thus "aggregate of points made by contestants in certain games and matches" (1742, in whist).
The sporting score-card is by 1877 (in cricket). The newspaper sports section score line is by 1965. Score-keeping in sports is by 1905. From the tavern-keeping sense comes the meaning "amount on an innkeeper's bill" (c. 1600) and thus the figurative verbal expression settle scores (1775; as cut scores, 1610s).
Meaning "printed piece of music" is recorded by 1701, said to be from the practice of connecting related staves by scores (in the "line drawn" sense). Especially "music composed for a film" (1927). In underworld slang, "money obtained in a crime," 1914. Meaning "an act of obtaining narcotic drugs" is by 1951.
The meaning "a cut, notch, scratch or line made by a sharp instrument," without reference to counting, is attested from c. 1400. By c. 1600 as "a line drawn."
also gas-light, "light, or a provision for light, produced by combustion of coal gas; a gas-jet," 1808, from (illuminating) gas (n.1) + light (n.). Used through the 19th and into the early 20th century as street and domestic lighting. Related: Gas-lighted; gas-lighting; gaslighting.
As a verb meaning "to deliberately make a person believe that they are insane," by 1961, perhaps 1956. This sense is from the 1944 film Gaslight, in which a 19th century woman (played by Ingrid Bergman, who won an Academy Award) appears to be going mad. It is later revealed that her criminal husband has been convincing her that she is insane in order to discredit her observations of his activities. Among the observable clues has been the dimming of their home's gaslighting whenever (as she later learns) the husband goes secretly into the attic: he has convinced her that she is imagining this, until a family friend sees it, too, which confirms the clue that uncovers the crime.
Brian: Tell me. Is there anyone else in the house now, except us and Elizabeth?
Paula: No. Why?
Brian: The gas just went down.
Paula: You saw that too!
Brian: Why, yes.
Paula: Oh, then it really happens! I thought I imagined it!
Brian: But all it means is someone else has turned it on.
Paula: Oh, no, no. I thought that too. But every night, I’ve been all over the house, there’s never been another light turned on. At last I can tell this to someone! Every night when my husband goes out…
Brian: …The light goes down?
Brian: And then what?
Paula: Well, then, I think I hear things. I watch and wait. Later on, the gas goes up again.
Brian: And he comes back?
Paula: Yes. Quite soon after. Always quite soon after.
The word seems to have received a boost in feminist literature in late 1970s.
… I had been told that my tonsillectomy was “not that bad” or that the dentist whose hands were between my legs was “fixing my teeth,” … My own favorite embodiment of this horror, still enjoyed by late-show insomniacs, is the 1944 film Gaslight, a tale which so impressed the public imagination that even today the word “gaslight” is used to describe an attempt to destroy another’s perceptions of reality and, ultimately, sanity itself. [Florence Rush, "The Best Kept Secret: Sexual Abuse of Children," 1980]
The sense evolved by 2016 to also mean "dismiss or discredit someone's viewpoint."