The modern term Sasquatch is a loan word (first used by Indian Agent J.W. Burns) based on the Halkomelem First Nations people's word sásq'ets.
In the myths of the Halkomelem people, what was sásq'ets? Were there identifiable stories told about it, or was it just a strange man-like hairy creature a friend of a friend caught a glimpse of in the woods, much like you hear about today?
Sasquatch: Legend Meets Science, by Jeff Meldrum, says (emphasis mine):
The various tribes across North America have attached their own names to the entity. These names number more than sixty, but most generally make reference to a "Wildman of the woods." In the 1920s, Canadian journalist J.W. Burns coined the term Sasquatch as a common denominator for the myriad of native names. Sasquatch derives directly from the word "sésquac." The original word, in the dialect of the Stó:lò dialect of the Halkomelem language, is used by the Coast Sasqu Indians of the Frase Valley and parts of Vancouver Island, British Columbia.
This translates "sésquac" as "wild man." This has a passage curiously similar to the one in Meldrum's book:
The term "sasquatch" is an anglicized derivative of the word "Sésquac", meaning "wild man". The original word, in the Stó:lõ dialect of the Halkomelem language, is used by the Coast Salish Indians of the Fraser Valley and parts of Vancouver Island, British Columbia.
Indian tribes across North America have a total of more than sixty different terms for the sasquatch.
Strangely enough, Meldrum is not listed in the "Useful Links" section, though that should not be confused with sources.
A fairly detailed list of name related to the creature is given by Ontario Sasquatch. Most related words are simply translated as "Sasquatch" (or "male Sasquatch" or "female Sasquatch"), though the word "Suhsq’uhtch", used by the Coast Salish people, is translated as "men, covered with dark fur, more than eight feet tall, who leave footprints about twenty inches long."
The Cowicham people gave the creatures the name "Thumquas.". Interestingly enough, the Cowicham people do speak Halkomelem, though Wikipedia describes them as speaking a branch distinct from that of the Stó:lò.
One of the most tantalizing sources I could find about the mythology surrounding the name was from the article "Retired principal puts out second pre-teen novel", by Clyde Woolman (Comox Valley Record) (emphasis mine):
It at least seems reasonably clear that the origin of the English word came from an Agassiz teacher in the 1920s who heard the term sasq'ets from the Sts'ailes (Chehalis) people) who live in the area, which described a sollicum, a being of the natural and supernatural world.
The word "sollicum" does not seem to appear anywhere else related the Chehalis mythology - at least, nowhere that I could find.
A page on SasquatchSighting.ca claims (emphasis mine),
The legends of Sasquatch have been part of the Chehalis tribe for centuries. In fact, their oral history says that sasquatch are their mortal enemies. And in their St:olo dialect the word Saskakaua roughly means “Place of the Wild Men”.
The Sunday Journal and Star says much the same thing. In the lower left-hand corner,
The existence of a troglodytic race inhabiting the mountains of British Columbia in many of the vast caves is a tribal legend among the Chehalis Indians and those of the Skwah Reservation, near Chilliwack, in the Harrison Lake district, about a hundred miles east of Vancouver. Among the Indians the race has been known for centuries by the name "Sasquatch," or hairy men.
The article later says that legends date back "at least three generations" (this was written in 1934).
This also says
The creature has a spiritual quality for the Chehalis--the spiritual and physical are closely connected in First Nations beliefs.
It also says that First Nations people all over have legends based around the creature, and that rock paintings appear to have been made of them.
I was not able to find anything about "Thumquas," unfortunately.
One a side note, the Quinalt people have a legend about a being known as "Glue-Keek". A Quinalt woman, Harvest Moon, tells the story (also given here:
Harvest Moon is familiar with several native legends about Bigfoot. The Quinault woman works as a storyteller at the Lake Quinault Lodge and other resorts along the coast during the summer.
She tells the story of the Glue-Keek monster she learned from a Lummi elder and its ties to the creation of mosquitos. The monster frightens tribal members and prevents them from hunting and gathering food.
His legs were as big as tree trunks, she said while swaying her hips and making arm and hand gestures during a recent storytelling presentation at the Lake Quinault Lodge. His skin was as tough as leather and his eyes had a hypnotic glow to them. The monster started chasing the women through the berry patch. He took his huge, big feet, knocking over every basket of berries, wasting them on the ground.
According to the legend, warriors from various tribes gathered and vowed to kill the monster. They dug a hole, tricked Glue-Keek into falling into it and burned him. As Glue-Keek perished, he swore he would return to drink the villagers blood. As his ashes ascended into the air, they transformed into mosquitos.
