My definition of traditional name is that name that appears in the stellar catalogs made in classical antiquity by Greeks and Romans and which were later translated and compiled by the Arabs to be later used again by Europeans from the Renaissance. I do not look for traditional Chinese or Mayan names, for example.
Most stars visible without a telescope were given names in Antiquity. These Greek or Babylonian names were transmitted and modified during the Middle Ages by Arabian compilers and many of them are still popular, such as Sirius or Vega.
Other names are longer and more "exotic", like Betelgeuse. I am thinking, for example, of really long and exotic names like Zubenelgenubi (Alpha Librae) or Zubeneschamali (β Librae). Are there any stars whose "ancient" or "medieval" name are longer?
Here is a list of arabian star names, later modified by europeans, but it may be incomplete, because greek catalogues included about 1.000 stars and this list only includes 200.