Boy:

The custom is to name a boy on the day of his circumcision, and  not beforehand.[1] [The name is given within the blessing that is said following the circumcision.] Even if the parents have already decided on a name, they should not call the child by the name until the actual naming after the Bris.[2]

Prior to birth:[3] Prior to the birth of the child, one is not to discuss with others the planned name of the child. [However, the parents may discuss it amongst themselves].

1. When to name if the child is sick and the Bris is delayed:[5]

One does not officially name the child until the actual circumcision, even if the child is sick and cannot be circumcised for many weeks or even years after the birth.[6] For the Mi Shebeirach that is said on his behalf, one should refer to the child by the term “Ben Hanolad or Yeled ben Plonis.”

Other customs and opinions: Some write that if the child is sick and is in need of prayer, then one may name the child by the first Torah reading opportunity, even prior to the passing of eight days.[7] However, when the circumcision takes place, the regular Mi Shebeirach for naming the child is to be recited.[8] Likewise, some write that one should not delay naming the child for more than 30 days, and in the event that the circumcision will be delayed for more than 30 days, then the child is to be named before his circumcision.[9] Others write that he should be named immediately after eight days have passed when the father is called up for an Aliyah, if the circumcision has yet to take place.[10] Others write that in the event one knows that the circumcision will be delayed, then in such a case he should be named immediately even prior to the passing of eight days.[11]

Informing the Rebbe to Daven for the child: Some write that in such cases the Rebbe would ask the individual to inform him of the intended name so he can pray for him to get better, although they should not yet publicize the name to others, not even his own parents.[12]

Pidyon Haben:[13] If the child will be having a Pidyon Haben prior to his circumcision which is delayed due to him being sick, then some write that the name should be given by the Pidyon Haben, although others maintain that even in this case the naming should be delayed until the circumcision.

No circumcision:[14] In the event that the boy will not be circumcised, such as if his brothers died due to circumcision, then according to all opinions he is to be given a name at the first opportunity of Torah reading after the birth.

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[1] Shut Hamaor 1:2134; Mivaser Tov 79; Chesed Leavraham Azulai Mayan 2 Nehar 52; Ziv Hasheimos 2; Bris Avos 8:1; Shevach Habris 20:1; Many Poskim in Nitei Gavriel 70:1 footnote 1

The reason: The custom is not to name the child until after the circumcision when the foreskin which comes from impurity has been removed only then can the holiness of the soul fall upon him. [Chesed Leavraham ibid]

[2] Shut Hamaor 1:2134; Mivaser Tov 79; Chesed Leavraham Azulai Mayan 2 Nehar 52; Ziv Hasheimos 2; Shevach Habris 21:1

[3] See Shut Hamaor 1:2134; Mivaser Tov 79; Chesed Leavraham Azulai Mayan 2 Nehar 52; Ziv Hasheimos 2; Chemdas Tzevi 5:42; Shevach Habris 20:1; Nitei Gavriel ibid

[4] Zecher David 1:1; Shevach Habris 20:31; Nitei Gavriel 71:2; See Igros Moshe 3:35

[5] See Nitei Gavriel 70:2-4; Shevach Habris 20:18-19; Toras Hayoeldes 60:6

[6] Chamudei Daniel Y.D. 262; Betzel Hachaochma 6:10; Zecher David Mamar 1:91; Sefer Habris p. 282:51; Nitei Gavriel 70:1 and 2; Shevach Habris ibid footnote 26 in name of Rabbanei Anash

[7] Az Nidbaru 13:73; Asia 4 p. 249 in name of Rabbi Moshe Feinstein and other rabbis and Admurim, brought in Nitei Gavriel 70:2 footnote 3 and Shevach Habris ibid

[8] Az Nidbaru 13:73; Otzer Habris 6:2-1 in name of Rav SZ”A

[9] Teshuvos Vehanhagos 1:604

[10] Koreis Habris 32; Bris Avos 8:2; Hillel Omer Y.D. 151

[11] Koreis Habris p. 298

[12] Shevach Habris ibid in name of Rabbi Label Groner; Nitei Gavriel 70:2 footnote 3

[13] Nitei Gavriel ibid

[14] Tiferes Banim 5; Sefer Koreis Habris; Mahram Brisk 2:7; Divrei Yatziv Y.D. 148; Nitei Gavriel 70:5; Shevach Habris 20:19 footnote 31