On both Naruto and Yu-Yu Hakusho, Kurama is the name of a powerful demon fox.
Does this have some meaning in Japanese mythology or folklore? Or did Yu-Yu Hakusho's author made it up, and Naruto's author "inspired" from it?
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The philosopher Hayashi Razan lists one of the three greatest of the daitengu as Sōjōbō of Mount Kurama. The demons of Kurama and Atago are among the most famous tengu.
The Naruto Wiki mentions this as a possible source of the name:
"Kurama" (九喇嘛) literally means 'nine lama'. Kishimoto was mainly inspired to create Kurama based on the character with the same name from the manga series Yū Yū Hakusho. The name may also reference Mount Kurama (鞍馬山), the sacred mountain said to be the home of the Tengu Sōjōbō who taught people ninjutsu and other Japanese martial arts.
The other one listed there, of course, is Kurama from Yū Yū Hakusho. This does appear to be cited, but the citation is to a book called the Second Artbook (this one, I think, pages 74-81), which I haven't been able to find online anywhere.
How it appears is that Yū Yū Hakusho chose this name in the early 1990s (probably from the name of the aforementioned mountain), then Naruto used this as inspiration for a similar character of their own.
Because of this choice from Naruto's mangaka, it seems likely that he also chose a similar character to use that name for. Kurama in Yū Yū was a yōko, it makes sense to reuse that name for a kitsune.
As for the original usage of the name Kurama in Yū Yū, one plausible explanation is that because of the name Kurama's connotation to the martial arts master Sōjōbō mentioned above, who possessed great wisdom and taught magic to Minamoto no Yoshitsune.
Stories depict [the kitsune] as intelligent beings and as possessing magical abilities that increase with their age and wisdom. Foremost among these is the ability to assume human form. (Wikipedia)
However, this does not appeared to be cited to any specific source, so it is also highly plausible that there is no direct connection between the use of the name Kurama and the fox.