The Rebbe's guidance on choosing a child's name and on a boy's first haircut at three:
The parents choose the name. Choosing a child's name is entrusted to the parents, who are given inspiration from Above for the name that suits the child's soul; for this reason the Rebbe did not usually suggest names himself, though he blessed the parents' choice (IK 1208, 1295, 1531, 2752, 3203, 4170, 4299, 6352, 6794). Both parents should consult together and agree (IK 1531, 4009, 4248). Where there is no fixed family custom about which parent names which child, alternating between them is suggested (IK 1332, 6613). If someone else already gave the name and it was announced with a Mi Sheberach, it should not be changed, though a name may be added (IK 2433).
Whom to name after. Naming a child after a tzaddik brings blessing, and the Rebbe expressed his pleasure when children were named after his father and after the Previous Rebbe (IK 1630, 7860). Take care when a name belongs to a living relative — without making a fuss, let the child's main name be a different one (IK 2144).
When to name. A baby girl is named at the first Torah reading after birth; there is no need to wait for Shabbos or for several days (IK 805, 862, 7641, 8452). We bless a newborn girl, like a boy, "to Torah, chuppah and good deeds" (IK 4028). A bris should take place as soon as the baby is healthy, without unnecessary delay (IK 2003). See also: IK 345, on the meaning of a name.
The upsherin: at three, and on time. A boy's hair and payos are not cut at all before he turns three (IK 1805, 7287). The first haircut is held on the third birthday itself — Chabad custom follows age three, not four — and should not be put off without strong reason, because it begins his chinuch and draws added holiness to him (IK 3384, 3438, 3946, 4979, 5081, 9318). Afterwards payos are left at the proper length and he wears a head covering, begins to wear tzitzis and to say blessings (IK 1233, 1242, 2803, 9334).
When the birthday is on a special day. If the third birthday falls on Erev Pesach, don't delay: cut the hair the night before or on Erev Pesach itself (IK 8335). If it falls during Pesach or Sefirah, halachah allows a haircut even on Chol HaMoed (IK 7287), but the Rebbe advised following the widespread custom and holding it on Lag BaOmer, in Meron if possible (IK 2091, 2539, 2586, 2624, 2671). If the birthday comes after Lag BaOmer, hold it on the birthday unless your family has a different custom (IK 9318). If it falls on Rosh Chodesh, hold it right after Rosh Chodesh (IK 6695, 8560); on Rosh Hashanah, hold it after the birthday (IK 8506); during the Three Weeks, wait until after Tisha B'Av (IK 9334). For other questions, consult the local Rav or experienced chassidim (IK 7420, 7529).
The celebration. The haircut itself should be on the birthday, but a public celebration can be held at any convenient time or place (IK 8532). There are no fixed instructions for the celebration beyond what the Previous Rebbe taught (IK 1125); the main thing is that it marks the start of the child's Jewish education (IK 939, 1189, 4403, 7930).