The Yalkut Shimoni on Mishlei (859) explains these names (and adds another, see at length):

Ari - "she'Hakol Misyarin mi'Menu" -- "that everyone is scared of him."

Kfir - "she'Kol Mi she'Ro'aihu Kofer b'Chayav" -- "whoever sees him denies his life."

Lavi - "she'Hu Chotef Levavos Shel Bnei Adam" -- "that he grabs hearts of people."

Layish - "she'Besaro Shel Adam Lash b'Shinav" -- "that the flesh of man he grinds with his teeth."

Shachal - "she'Chalim mi'Panav" -- "that they are sick from before him."

Shachatz - "she'Hu Chatz b'Shinav" -- "that he cuts with his teeth."

I assume that the simple explanation behind the Yalkut Shimoni is that these names were used in order to express different ideas regarding the lion, each best fitting the Pasuk where it was used.

All the best,

Yaakov Montrose