TT #20 · Parshas Tazria · Aliyah 1 · Passuk 3
English: On the eighth day the flesh of his foreskin shall be circumcised.
Chabad.org — Chumash, English translation + Rashi for Vayikra 12:3
Torah Temimah cites Niddah 31b.
E-Daf — Niddah 31b · Shas.org · Steinsaltz Center — vowelized Niddah 31b
English: Why did the Torah prescribe circumcision on the eighth day? So that everyone would not be rejoicing while the child’s father and mother are distressed.
Reason supplied by Chazal
The Sages explain the eighth-day timing as a way to avoid leaving the parents distressed while everyone else rejoices at the circumcision.