It goes without saying that there are also various other tags within this franchise such as yuu-gi-ou gx and yuu-gi-ou zexal. If this is accepted, it would affect the other tags as well. I can create the BUR, but first I'd like to see if this is OK. I will also volunteer to edit the affected wikis.
I think it should be yuugi ou, because these are two Japanese words: "yuugi" (game) and "ou" (king). That is "king of games". This is assuming we want to use our own romanization system that reflects the original Japanese as closely as possible.
It is a well-known fact that the official romanization is "Yu-Gi-Oh!". I know folks are used to it so it should remain as an alias at least, but that does not make much sense in Japanese. In my opinion, "Yu-Gi-Oh!" looks like three random gibberish syllables instead of a phrase meaning "king of games".
To be fair, we could use yu-gi-oh! instead, because that's a well-known romanization. Personally, I don't like "yu-gi-oh!" because it messes with the Japanese meaning, but maybe it would be helpful for some people.
Either way, the current yuu-gi-ou is a poor choice. It's not the official romanization and it does not very accurately reflect the original Japanese meaning. That yuu-gi-ou is a mess based on both romanization systems.
Related discussion: topic #15900 (from 2019).