1) Naming A Boy– It is well known that the custom is to name the boy at the Bris. This custom is sourced in the Rishonim. (Siddur Ri Ben Yakar) The reason for this is that only after the baby is in his perfected state, after the removal of the orlah (foreskin), is the baby prepared to receive his Jewish name. (Chesed l’Avraham 2:52)
2) The poskim discuss what to do if a child is ill and will not be able to have the bris on time at 8 days: A) The Sefer Chemudei Daniel is of the opinion that if a child is ill and will not have a bris for weeks, one may give him a name before the bris. According to the Chamudei Daniel one should name the baby before he is 8 days old, while the Sefer Kores Bris (page 25) maintains one should name before the bris, but after the baby is 8 days old. Indeed, there were rabbanim who endorsed naming an ill baby before the bris so that the baby will have a name that others can use when davening for his recovery. Rav Moshe Feinstein zt”l and the Satmar Rebbe zt”l are also cited as advising one to give an ill child a name, even before the bris. (Refer to Kovetz Asia page 244) B) Harav Shach zt”l is cited as ruling that one should only name the baby before the bris if the baby is in a life threatening situation. However, if the bris is postponed because the baby’s bilirubin levels are high, however, the baby is not in a life threatening situation, one should not name the baby before the bris. C) The Steipler Gaon zt”l is cited as ruling that one should never name the baby before the bris. According to the Steipler Gaon zt”l if one wishes to daven for the baby one would pray for “tinok (child) ben plonis.”
3) Naming A Girl– The custom is to name the girl when the father receives an aliyah to the Torah. The Gabbai recites a special Mi Sheberach and the baby is formally named.
4) There are various customs as to when to name the baby girl, however, there are primarily two approaches:
A) The view of the Bnei Yisasschar was to name the baby at the first Torah reading after the birth, even if that is a Monday or Thursday, and one does not wait for Shabbos. (See Minchas Yitzchak 4:107) This is also the view of the Minchas Elazar of Munkatch zt”l and is followed by Munkatcher Chassidim. This is also the custom of Chabad, (Shaarei Halacha U’Minhag Y.D. page 297) Ziditshov and Spinka Chassidim. (Netai Gavriel Niddah vol. 3 page 595)
Similarly, according to the custom of Chabad Chassidim, if a mother gives birth on Shabbos morning after Shachris, the father will name the baby at Mincha.
B) While others name the baby at the Torah reading on the first Shabbos after birth. As it states in the Sefer Ben Uziel Parshas Shemos, “I have heard in the name of Rav Yechezkal of Shinova zt”l that he was particular not to name a girl during the week, rather only to do so on Shabbos. He explained that the reason being that a baby boy receives his holy neshama at the bris milah, however, a baby girl receives her holy neshama on Shabbos.” The Avnei Nezer is also as cited as being very particular that one name a girl on Shabbos. (Siach Sarfei Kodesh 3:283) This is also the practice of many chassidim, including Siget, Satmar, Tchernobel, Sanz, Ger, Amshinov, Bialeh and Belz. This is also the custom of many sefardim. (Mishnas Yehoshua Bas page 209)
5) The custom of Belz is that they name on Shabbos. However, if the baby is born on Friday, they will wait and name the baby the following Shabbos (8 days later).
6) The custom of Sanz and Babov is to name the baby on Shabbos. This is true even if during the week is Yom Tov, they still wait for Shabbos.
7) Harav Gavriel Zinner shlit”a writes that all agree that if a baby girl is born ill and requires others to daven for her health that one should name her right away and one need not wait for a day that the Torah is read.