Presumably, you are referring to the colloquial usage of the term “Kiddush” – a celebration involving food, which ideally also involves words of thanks to HaShem and/or divrei Torah. This term is borrowed from the communal “Kiddush” held (usually) in shuls after the tefilla of Shabbat morning, which consists of a small meal prefaced by the Shabbat Kiddush. (As we will see below, there may actually be some significance to scheduling the Kiddush upon the birth of a girl to take place after the Shabbat morning tefilla.)
In general, it is a mitzva to thank HaShem for joyous and/or miraculous events. One of the applications of this concept is Birkat HaGomel, which we recite after being saved from danger.[1] While there is no requirement to have a meal in addition to the recitation of this beracha, sources indicate that it is a nice idea;[2] such a meal would parallel the eating of the Korban Toda[3] in the time of the Beit HaMikdash.[4]
It is unclear whether one should have a celebratory meal in honor of the birth of a child. After a boy is born, there is a seuda at the time of the brit mila, and there is another one at the pidyon haben (when applicable). However, these meals celebrate the fulfillment of those mitzvot; while they are normally held not long after birth, they are not celebrations of the birth per se.
The gemara[5] mentions two celebrations upon the birth of a boy, shevua haben and yeshua haben. While some explain that these terms refer to the celebrations held after the brit and the pidyon haben,[6] there are opinions that one of them is what Ashkenazim call a shalom zachar.[7] Although it might appear that a shalom zachar is certainly a celebration of the baby’s birth, this is unclear. There are three basic approaches regarding the purpose of a shalom zachar: 1) it is a celebration of the safe extrication of the child from his mother’s womb at birth.[8] 2) It is held to console the baby for the loss of the Torah learned during gestation.[9] 3) It is related to the upcoming brit.[10] The Dagul Merevava[11] argues that the shalom zachar must be related to the brit mila rather than to the birth itself, because if it were related to the birth then perforce there would be a similar event upon the birth of a girl. For this reason he says that the shalom zachar should be held the night before the brit, which is indeed the Sephardic practice (called a brit Yitzchak). Many Ashkenazim also have special observances on that night (in Yiddish, vach nacht). If so, it is unclear whether even a shalom zachar is a birth celebration.
There is a Sephardi minhag of a zeved habat, which normally includes giving her a name.[12]
For Ashkenazim, there is no set time or formula for the celebration of a girl’s birth, and the consensus is that one is not obligated to have such a celebration. In some circles, people try to have some celebration on the day of her naming; this naming is normally done as part of a Mi Sheberach in conjunction with Kri’at HaTorah (and is an event to which some attribute tremendous meaning[13]). Some name a baby girl specifically on Shabbat, when typically more people are present.[14] (The presence of many people is generally desirable for thanksgiving to HaShem[15]). This common practice of naming a girl on Shabbat and then celebrating immediately after the davening may explain why people commonly call such an informal meal a “Kiddush” no matter when it takes place.
There is a baraita[16] which refers to a shevua habat as a female parallel to the shevua haben (see above). This could serve both as a source for the practice of a celebration upon the birth of a girl, as well as being a possible reason for having it sometime on Shabbat. The gemara[17] discusses Boaz making 120 celebrations in honor of his children, and Rabbeinu Gershom[18] says that 30 of those celebrations were held after the births of his 30 girls.
The truth is that while there is no specific obligation, timing, or setting, for such a celebration, simple logic dictates that one should thank HaShem for such a monumentally joyous occasion.[19] In addition, some stress the importance of the berachot people give the baby and parents at the celebration,[20] and there are some “legends” that attribute great importance to this.
In summary, there is abundant basis for a “Kiddush” in honor of the birth of a girl being appropriate, and there are some sources that imply an established practice on the matter. It is our opinion that making such a Kiddush should not become a necessary burden on the heads of new parents, by having them feel they must do it at a specific time or a specific way. It certainly should not necessarily include large amounts of expense or toil, which may cause the parents difficulty at an often stressful, even if joyous, time. Hopefully, the parents will be able to find an appropriate, and not too challenging, opportunity to publicly share their joy and gratitude to HaShem.