In this particular chapter, we present only customs involving the naming of newborn children, not adopted children, converts or a sick person who's name is being changed.
Introduction
In general, the giving of a name should be looked upon as a great responsibility that involves serious consideration by the parents. In many places in Kabbalah and Chassidus it is explained how the name by which something is called is the conduit through which its life-force is infused. Therefore, the AriZal (Rabbi Isaac Luria) writes. "When a person is born and his father and mother give him his name...the Holy One puts into their mouth the particular name required for that soul" (Sefer HaGilgulim, introduction 23; Emek HaMelech, Shaar 1 end of Ch. 4; Or HaChayim on Devarim 29:17)
Both Parents Should Agree
The giving of the name should be with the agreement of both parents together, as it is stated in various sources. The parents may, however, allow someone else to give the name, acting as their agent. If in fact it was not done this way (and instead the name was given by the grandmother or the like without the parents' permission) the parents can decide on the particular name they wish. If a name was already formally given in front of a Sefer Torah, the name should not be canceled (G-d forbid), but another name of the parents choice may be added.
When the Parents are in Disagreement
The Rebbe wrote to someone, that in places where there is no established custom, the names should be given in alternating order: the first to the father, the second name to the mother, the third again to the father, etc.
In Scriptures we occasionally find that the children were named sometimes by the father and sometimes by the mother. For example, regarding the tribes it is written "and she called his name..." In fact, all the names of the tribes (except Binyamin) were given by their mothers. We can draw no inference from this, as it makes no sense to say that the names of all sons belongs to the mother and the father has no part in it, especially since we do find in Scriptures children named by their father, and occasionally even by both together (Yishael, Binyamin) or even by outsiders (Zerach, Peretz, Oved).
[Possibly the reason why the tribes were named by Rachel and Leah is in keeping with the teachings of the Sages (Beraishis Rabbah, end of 72) "the Matriarchs were prophetesses and the Patriarchs were inferior to them in prophecy" as it says, "everything that Sarah says to you, hearken unto her voice." Also, the names of Yitzchak and Esav were apparently given by Sarah (Beraishis 21:6) and Rivkah (Rashi, ibid. 25:26).
Naming After a Living Person
Generally speaking, the custom differs between Sephardim and Ashkenazim. Among Sephardim, naming the child after the living grandfather is regarded as a great honor for him. Thus, he names his son after his father while he is still living. Ashkenazim are scrupulous not to name the child after a living grandfather, rather, only after someone who has already passed on to the next world.
A certain Jew once asked the Rebbe, how it is possible that he replied to someone (from a Sephardic community) that he could name his son after his father who was still living, when in fact the Ashkenazim are careful not to do this? The Rebbe answered him: "How can I ignore such a custom, when it is related explicitly in Chumash (end of Parshas Noach) that Terach named his son after his father who was still alive, as we calculate from the numbers stated in those verses?"
Naming a Grandchild with a Grandparents' Name
According to Ashkenazic custom, if a child has been given more than one name and afterwards it was remembered that the child's living grandparent bears one of those names, the child should only be called by the one non-shared name. The same applies to the English name.
Naming After Tzaddikim
Many have the good custom of naming their children after the holy and the righteous - tzaddikim and tzidkaniyot.
Names given after tzaddikim and tzidkaniyot should preferably not be combined with other names.
The Proper Time for Naming a Boy
The proper time to name a baby boy is during the Bris ceremony. This includes naming at a delayed Bris. However, if the child is sick and there is a need for a name to be able to pray on the child's behalf (or in the case of a firstborn whose Bris is after his Pidyon Haben) a name can be given at an earlier time.
The Proper Time for Naming a Girl
There are various customs regarding the naming of a baby girl: at the first Shabbas Torah reading or at the Shabbas Torah reading following the fifth day of birth, or at the first Torah reading after the birth. A Jewish girl is regarded as one who is born circumcised (Talmud Avodah Zarah 27b) and thus her Jewish soul enters immediately upon birth; therefore, she should be named as soon as possible.
The Rebbe writes in Igros Kodesh Vol 14, p. 56 "Regarding the proper time for naming a girl - we know of a ruling by my saintly father-in-law the Rebbe, following the actual practice of the Mitteler Rebbe, that it should be done at the first occasion of which the Torah is read..."