Origins
The word 'point' is one of the most polysemous words in the English language, with the Oxford English Dictionary recording over seventy distinct senses. All of them trace back to a single Latin image: the mark left by something sharp piercing a surface.
The word entered Middle English in the mid-thirteenth century from two related Old French forms: 'point' (a dot, a mark, a puncture, a small amount) and 'pointe' (a sharp tip, the end of a weapon). Both descended from Latin 'punctum' (a prick, a point, a tiny hole), the past participle of 'pungere' (to prick, to pierce, to sting), which itself traces to the PIE root *pewǵ- meaning 'to prick' or 'to stab.'
The Latin verb 'pungere' was exceptionally fertile in English through both French and direct Latin borrowings. 'Puncture' (a pricking) came directly from Latin. 'Punctual' originally meant 'of or relating to a point' before acquiring its modern sense of 'arriving at the appointed point in time.' 'Pungent' (stinging, sharp-smelling) preserves the original sensory quality of the verb. 'Compunction' (a pricking of the conscience) combines 'com-' (intensive) with 'pungere.' 'Poignant' (piercing to the feelings) came through French from the same root. Even 'punch' (the striking action) is likely from a variant of the same Latin family.
Figurative Development
The semantic expansion of 'point' from 'sharp tip' to its modern range of meanings proceeded through a series of logical metaphorical steps. A 'point' as a sharp tip led to a 'point' as a dot or mark (the mark a sharp instrument makes). A dot on a surface became a location in space (a 'point' on a map). A location in space became a moment in time (a 'point' in time, 'at this point'). A single mark or unit became a unit of scoring (to score a 'point'). A single item in a series of marks became an item in an argument ('my next point'). The essential item in an argument became the essential meaning ('the point of the story'). Each step is a short metaphorical leap, but the cumulative distance from 'sharp tip' to 'essential meaning' is enormous.
The word 'appoint' (originally 'to settle to a point,' to fix precisely) and 'disappoint' (originally 'to remove from an appointed position') further illustrate the word's metaphorical reach. 'Disappoint' shifted from its concrete sense of dismissal to its modern emotional sense — the feeling of having one's expectations undone — during the sixteenth and seventeenth centuries.
In mathematics, 'point' acquired its technical geometric meaning (a location with position but no dimensions) through the influence of Euclid's 'Elements,' where the Greek word 'sēmeion' (sign, mark) was translated into Latin as 'punctum.' This mathematical sense — the most abstract refinement of the original 'dot' meaning — has been standard since the medieval period.
Later History
The phrase 'to point' as a verb (to direct attention, to indicate with the finger) developed in the fourteenth century from the noun. 'Pointer' (something that points) followed naturally. 'Pointless' (lacking a point, whether a sharp tip or a purpose) dates from the sixteenth century, elegantly exploiting the word's double meaning of 'sharp end' and 'purpose.'
Phonologically, the word has changed little since its adoption from French. The Old French diphthong 'oi' was pronounced /ɔɪ/ in Anglo-Norman French, close to its modern English pronunciation. The final 't,' often silent in modern French, has always been pronounced in English. The word's brevity and phonological stability have contributed to its extraordinary utility — a single syllable that can be deployed in virtually any context where location, purpose, argument, or sharpness is at stake.