The name "jigsaw" came to be associated with the puzzle around 1880 when fretsaws became the tool of choice for cutting the shapes. Along with fretsaws, jigsaws and scroll saws have also been noted as tools used to cut jigsaw puzzles into pieces.
The term "jigsaw puzzle" dates back to 1906.
Puzzle historian Anne Williams attributes the 1907-1910 puzzle craze to a mix of factors: a poor U.S. economy in 1907, increased participation of middle- and upper-class women in arts and crafts activities like puzzle making, and the introduction of the foot-powered scroll saw. Jigsaw puzzles also soared in popularity during the Great Depression, as they provided a cheap, long-lasting, recyclable form of entertainment. It was around this time that jigsaws evolved to become more complex and appealing to adults.
(Wikipedia, Jigsaw Puzzle)