The saxophone is one of the few instruments with its maker's name on every one.
The inventor was Antoine Joseph Sax, called Adolphe, a Belgian instrument maker who lived from 1814 to 1894. Etymonline says he devised the instrument around 1840. Wikipedia says he was born in Dinant, was first based in Brussels, and moved to Paris in 1842 to set up his business.
His goal was specific. According to Wikipedia, Sax wanted an instrument with a brass instrument's projection and a woodwind's agility. He combined a single-reed mouthpiece like a clarinet's with a conical brass body. Etymonline describes the result as a metal, reed-played instrument meant to be a more resonant substitute for the clarinet in military bands.
The legal protection came later. Wikipedia reports that Sax received a 15-year patent on 28 June 1846, covering 14 instruments in two groups of seven. The development date is looser: the same article says around 1840 in one place and the early 1840s in another, so the exact start is approximate.
The name is plain construction. The French word saxophone combines the surname with -phone, which Etymonline says comes from Greek phone, "voiced, sounding." Etymonline records the English word in 1851. The surname itself is a variant of Sachs, which means "Saxon."
Sax's father was an instrument maker too. Etymonline says Charles Joseph Sax invented the saxhorn in 1844, a trumpet-family instrument also made for military bands. So the family name was already on brass before it landed on the saxophone.
The instrument spread quickly in military circles. Wikipedia says the Garde Republicaine band used eight saxophones by 1856. The family of names grew with it, and Etymonline dates soprano saxophone from 1859 and alto from 1869.
Nicknames came later. Sax as a shortening is attested by 1923, and by 1955 axe was jazz slang for the saxophone before it broadened to other instruments.