Sheilos u'Teshuvos Maharshag 3:5 notes that his son once asked him why the recitation of the Berachah on the new moon is called Kiddush Levanah - exactly what kind of Kiddush is indicated by this Berachah? The Maharshag suggested that the name is, in fact, a typographical error, that initially the Siddurim said Chiddush Levanah - "The Renewal of the Moon" - indicating that the Berachah was to be recited each month when the moon started providing light again.
Minhag Yisroel Torah (426:1), however, finds it hard to believe that this typographical error was repeated so many times - although, in fact, in Shas and in the Rambam and other Rishonim the term Kiddush Levanah does not, in fact appear. After all, the term does appear in the Darchei Moshe, Beis Yosef and Rama. He therefore suggests that the term is used to commemorate the Kiddush HaChodesh - the recitation of the word Mekuddash (Sanctified) by the Sanhedrin at the time that the calendar was set by them each month in turn.
Perhaps we can suggest that the terminology Kiddush is used because the Berachah is extended to include a supplication that Hashem restore the moon to its full size and glory, as it was before it was diminished. It is therefore not a blessing on the moon, but a blessing to the moon. This property of the extended Berachah is underlined by the word Kiddush, in the sense of uplifting and elevating something mundane - in this case, our tefillah is that the moon should be elevated and restored to its ideal state.