c.1200, "minute amount, single item in a whole; sharp end of a sword, etc.," a merger of two words, both ultimately from Latin pungere "prick, pierce, puncture" (see pungent). The Latin neuter past participle punctum was used as a noun, meaning "small hole made by pricking," subsequently extended to anything that looked like one, hence, "dot, particle," etc. This yielded Old French point "dot; smallest amount," which was borrowed in Middle English by c.1300.
Meanwhile the Latin fem. past participle of pungere was puncta, which was used in Medieval Latin to mean "sharp tip," and became Old French pointe "point of a weapon, vanguard of an army," which also passed into English, early 14c.
The senses have merged in English, but remain distinct in French. Extended senses are from the notion of "minute, single, or separate items in an extended whole." Meaning "small mark, dot" in English is mid-14c. Meaning "distinguishing feature" is recorded from late 15c. Meaning "a unit of score in a game" is first recorded 1746. As a typeface unit (in Britain and U.S., one twelfth of a pica), it went into use in U.S. 1883. As a measure of weight for precious stones (one one-hundredth of a carat) it is recorded from 1931.
The point "the matter being discussed" is attested from late 14c.; meaning "sense, purpose, advantage" (usually in the negative, e.g. what's the point?) is first recorded 1903. Point of honor (1610s) translates French point d'honneur. Point of no return (1941) is originally aviators' term for the point in a flight "before which any engine failure requires an immediate turn around and return to the point of departure, and beyond which such return is no longer practical."