In the volume 11, we see Watanuki naming the bird he gave to Himawari, and this makes the pipe fox jelous and sulking because Watanuki didn’t name it, as he did with Tanpopo. He at first says that he can’t name it because well, the pipe fox isn’t his, it’s Yuuko’s, but the woman answers him:

“Names are very important things. Didn’t I say that at the very beginning? And for that very reason, one wants to be give a name by the person one loves the most.”
- Volume 11, pp. 41-42

We see a very excited pipe fox watching Watanuki after this was said and so the boy finally decides to actually naming it saying:

“Mugetsu. When you get big, your eyes turn into very moon-like crescents. When they’re like that, I can hardly tell where they are anymore. It’s like the moon isn’t there. And so, Mugetsu.”
- Volume 11, p. 43

The small fox is so happy that it hugs Watanuki so hard that he can hardly breath.

The translation of the name Mugetsu is “no moon” or “moonless”. The characters used for the name are – the second kanji is the one for the moon, while the first would be nothingness. The word mugetsu is used for moonless sky in poetic way in Japanese language.