"The cougar has numerous names in English, of which puma and mountain lion are popular. Other names include catamount, panther, painter because of its black tail tip, and mountain screamer. In North America, "panther" is used most often to refer the Florida panther sub-population. In South America, "panther" refers to both the spotted and black color morphs of the jaguar, while it is also broadly used to refer to the Old World leopard.
The cougar holds the world record for the animal with the highest number of names due to its wide distribution across North and South America. It has over 40 names in English alone.
"Cougar" is borrowed from the Portuguese çuçuarana, via French; the term was originally derived from the Tupi language. A current form in Brazil is suçuarana. "Puma" comes, via Spanish, from the Quechua language of Peru."
Source and further information:
https://en.wikipedia.org/wiki/Cougar
Some other names:
"The names "Erie" and "Eriez" are shortened forms of "Erielhonan," meaning "long tail." The Erielhonan were also called the "Cat" or the "Raccoon" people."
https://en.wikipedia.org/wiki/Erielhonan
"In North America, the Spanish call the lion léon de montaña, the Mayans cabcoh and the Guianans tig rouge. Texans also have names for their lions, including puma as in its taxonomic designation Puma concolor, cougar, panther, and catamount, meaning “cat of the mountain.”"
http://www.tpwmagazine.com/archive/2005/jun/legend/