Antoine Godin (also referred to in some sources as Thyery Goddin), was an Iroquois who was in what would become Idaho with Donald Mackenzie’s fur hunters. They were looking for whatever might grow fur, particularly beaver, for the North West Company out of Montreal. Godin, or Goddin, found what we today call Lost River. He was impressed enough with the area that he gave the valley and river his name, i.e., Godin River and Godin Valley.
According to Lalia Boone’s well-researched book Idaho Place Names, a Geographical Dictionary, when Godin later returned to the area, he couldn’t find the river, thus the name Lost River became popular. Of course, either explanation could be describing the same phenomena.
Thanks to Godin’s discovery, the river was trapped out of beaver by 1824.