Santa Claus is a Dutch name that English speakers heard and slowly wore smooth.
The root is Saint Nicholas, a Greek Christian bishop of Myra in the 4th century. Myra is now Demre, in Turkey. Wikipedia reports that he was remembered for generous gifts to the poor, including dowries for three poor daughters. He became a patron saint of children, among other groups.
In the Netherlands, devotion to him became the folk figure Sinterklaas. The name is a contraction of Middle Dutch Sinter Niklaas, which simply means Saint Nicholas. Etymonline traces the English word to a Dutch dialect form, Sante Klaas.
Dutch settlers carried the customs to New Amsterdam in the 17th century. The English took the city in 1664. Wikipedia notes that in 1773 a New York newspaper referred to St. Nicholas as "otherwise called Santa Claus." Etymonline records the name that year as "St. A Claus" in the New York Gazette, the earliest use it knows.
The shape of the figure came later. Livius.org says Washington Irving's 1809 satire of New York history invented much of the later tradition, portraying the saint as a stout, pipe-smoking Dutch citizen. The look we now call Santa was assembled over the 19th century, not by one author or advertiser.
The short form "Santa" is recorded from 1874, per Etymonline.
Santa has cousins that are not the same thing. Kris Kringle comes from the German Christkindl, "Christ child." Wordorigins finds the spelling in a Philadelphia ad of 1841, after German immigrants brought the custom. Father Christmas is an older and separate English figure, recorded from the 1650s, who later merged with the saint.
So the name travels from a Greek bishop to a Dutch contraction to a New York newspaper. Each step is a small sound change, which is how Saint Nicholas became Santa.