At first glance, I thought that river is called snake due to many snakes in that river. But then, i am totally wrong, lol.

The explorer Wilson Price Hunt of the Astor Expedition named the river as Mad River. ... Eventually, the name Snake River was derived from an S-shaped gesture the Shoshone tribe made with their hands to represent swimming salmon. Explorers misinterpreted it to represent a snake, giving the river its present-day name.
^Source