We make a bris on the eighth day. A korban has to be eight days old before being brought on the mizbeach. The new moon according to many has to be eight days old from when its 'born' again till me make a brocho. What is the siginificance of the eighth day from birth, they must all have a similar reason. Although with a bris a reason is given that the mother who then becomes tahor can also join in, or becoming 'lmaalo', this cannot apply to the other cases, the moon is already 'lmaalo' at 'birth' and why should an animal become 'lmaalo'.There are many instances of 'eight' in the Jewish religion as other posts here have shown, and they could all have varied reasons. This question is pertaining specifically to eight days from birth.
There is a Midrash (Vayikra Rabba 27:10) that says:
דבר אחר: והיה ז' ימים תחת אמו. ר' יהושע דסכנין בשם ר' לוי אמר: משל למלך, שנכנס למדינה וגזר ואמר: כל אכסנין שיש כאן לא יראו פני, עד שיראו פני המטרונא תחלה. כך אמר הקב"ה: לא תביאו לפני קרבן, עד שתעבור עליו שבת שאין ז' ימים בלא שבת ואין מילה בלא שבת, הדא הוא דכתיב: ומיום השמיני והלאה ירצה.
Without fully translating it, R' Levi is addressing why a Korban is not brought until the 8th day. He brings a parable of a king who decreed that before one may visit him, they must visit the queen. So too, G-d said that one should not bring a sacrifice until that animal has seen a Shabbat (Since after 7 days a Shabbat has definitely passed, no matter when the animal was born). So too, one does not perform a circumcision until the child has seen one Shabbat.
The Chatam Sofer (Teshuva 102) quotes someone who connects this idea to waiting until after the 7th day from the Molad to say Kiddush Levana, although he questions it. See here for an analysis of the Teshuva.