Yeah, Inwas wondering this for a long time. The two exceptional carriers that don't have to do with the modern Imperial Japanese Carrier fleet naming convention (that is a mouthful) are Kaga and Akagi.

Good examples of the modern naming convention for Japanese carriers are the following...
Shokaku (Soaring Crane) and Zuikaku (Auspicious Crane)...

Hiryu (Flying Dragon)...

Taiho (Great Pheonix)...

But my question is what other names could the Japanese come up for carriers?

One that I know that of is from world of warships the name of the tier ten Japanese carrier translate to white dragon in English though I dont know how it's said in Japanese

5463671
The Carriers were often named after flying creatures such as dragons and birds: Akagi, Kaga, and Shinano were exceptions since Akagi was originally suppose to be a battlecruiser, and Kaga and Shinano were originally suppose to be battleships. There were other exceptions, such as ships that were converted or in the case of Katsuragi and Amagi completed with parts intended for use in destroyers and cruisers

5464217 Okay.