A silhouette is named after a man of whom no portrait survives. Étienne de Silhouette was painted in oils, but Wikipedia reports that those pictures were destroyed during the French Revolution. Not even a silhouette of him is known.
He was born in Limoges in 1709. In March 1759, in the middle of the Seven Years' War, Louis XV made him controller-general of finances. His job was to rescue a treasury drained by the war. He cut royal household spending, trimmed state pensions and proposed a tax on visible signs of wealth, such as doors, windows, servants and luxury goods. In October he ordered gold and silver tableware melted down. By November 1759 he was out, after about eight months.
His economies made him a figure of fun. Things that looked skimpy or cut-rate were mockingly described as à la Silhouette, in the Silhouette manner. The cheapest way to get a likeness in those days was a black profile cut from paper or card. It cost a fraction of a painted miniature. The name attached itself to these profiles and outlasted every other joke about him.
That is the usual explanation, and French sources gave it by 1766. It is not the only one. Etymonline notes that some theories tie the word instead to the shortness of his time in office. Another story says he cut paper profiles himself at his château in Bry-sur-Marne. That version is often repeated but cannot be proved, since the château burned in 1871, according to Word Histories.
English took the phrase from French. Etymonline records "portrait à la silhouette" in 1766 and plain "silhouette" for a black profile portrait in 1792. Wikipedia notes that people in the 1700s mostly called these pictures "shades" or "profiles." The new word only became common in the early 1800s. From 1843 it could mean any dark outline seen against light, and by 1901 the shape of a fashionable garment.
The surname itself is a French spelling of a Basque name. His father came from Biarritz, and the Basque form Zuloeta seems to contain zulo, "hole," possibly meaning a cave. So the word for a shadow may trace back to a place full of hollows.