When one should name a daughter;
Why some gedolim insist on traditional names;
Why the naming is such an important ocassion, and others.
Here is part one.
When should you name your daughter?
a. A Lubavitcher Journal brings a minhag from the sefer Bris Avos to name her immediately, no need to wait for K'rias HaTorah, no need to wait for Shabbos, just do it the day she's born.
b. The Minchas Yitzchak brings from the author of the Darkei Teshuva that the parent should name the daughter at the very first K'rias HaTorah after she is born.
c. Reb Moshe Feinstein held that the naming should not be within the first three days after the birth, and then named at a mi shebeirach at a K'rias HaTorah. That is the minhag of the Feinstein family, with the exception of a certain Moshe Eisenberg, (named after his Great Grandfather Reb Moshe), who named his daughter the day after she was born. Another Feinstein Hanhaga is to know the difference between big things and little things, and not to lose any sleep over little things. I think this is a relatively little thing, around the same level as breaking a plate at the T'naim or whether you stand or sit when you say Lecha Dodi (and when you turn around for Bo'i b'shalom) I know that some people are obsessive about "their minhag" no matter what. I remember a bachur that came over to me in yeshiva with a terrible problem. His minhag is to stand for Lecha Dodi, and the yeshiva sits. He was faced with a terrible dilemma. What should he do???? I told him that he doesn't need to be mattir neder, and while he's in yeshiva, he should do what everyone else does.
d. Some say no sooner than the fifth day, unless the fifth day is preceded by a Shabbos, in which case you can name her on Shabbos.
e. Bostoner and Karliner Chasidim name only on the second Shabbos, based on the Medrash (brought by the Taz in YD 265 SK 13) that Kedusha cannot be imparted to a living thing before it experiences a Shabbos- before it appears before the Matron.
f. The Ya'avetz, in his Siddur, says that the name should be given on the first Shabbos that the mother comes to shul, and he doesn't qualify it by any number of days after the birth. Alternatively, the father should name her in shul after four weeks (28 days.)
g. After 30 days.
h. There's a famous opinion from a forebear of the Chida that you should name a girl no sooner than forty days after her birth (discussed, and in a clearer print, here.)
So, to sum it all up, the first day, the third day, the fifth day, at the first Krias HaTorah, on the second Shabbos, on the first Shabbos the mother comes to shul, after 28 days, after 30 days, and after 40 days. I'm glad I was able to clear this up for you.
More seriously: Please don't assume parity among the authors of the opinions cited above. The recorders of these varied minhagim comprise an extremely wide range of authoritativeness, from nobodies to Geonim. In practice, one should follow the minhag of the place or kehilla to which the parents belong. The name of a Jewish child is an element of his or her relationship with the community, and one should follow the minhag of the community one is a part of. As the Ramban says in Bamidbar 1:32,
הבא לפני אב הנביאים ואחיו קדוש ה' והוא נודע אליהם בשמו יהיה לו בדבר הזה זכות וחיים, כי בא בסוד העם ובכתב בני ישראל וזכות הרבים במספרם
Next installment: Why the Naming of a Child is a Profoundly Important Event.