The Shulhan Aruch writes (Orah Haim 223:1; listen to audio recording for precise citation) that when a woman delivers a baby boy, both she and her husband should recite the Beracha of "Ha’tob Ve’metib." According to the Shulhan Aruch, this Beracha should be recited in full, with the phrase "Hashem Elokenu Melech Ha’olam." The Mishna Berura (commentary by Rabbi Yisrael Meir Kagan, 1839-1933) explains that the word "Ha’tob" refers to the father’s joy and gratitude, and "Ha’metib" gives praise for the mother’s feeling of joy and gratitude. Some have explained that the parents experience unique joy upon the birth of a boy because male children inherit their parents’ possessions after their passing. The birth of a boy thus ensures the perpetuation of the parents’ legacy, for which they experience immense joy, warranting the recitation of this Beracha.
In any event, although this is the view of the Shulhan Aruch, the Ben Ish Hai (Rabbi Yosef Haim of Baghdad, 1833-1909) cites from the Hesed La’alafim (Rabbi Eliezer Papo, 1770-1828) that the common custom does not follow this position. Rather, it is customary for the father and mother to rely on the Beracha of "She’heheyanu" that the father recites during the Berit Mila. When the father recites this Beracha, both he and his wife should have in mind for the Beracha to refer not only to the Misva of circumcision, but also to the occasion of their son’s birth.
This is, indeed, the accepted Halacha. Therefore, despite the fact that the parents undoubtedly experience immense joy at the time of the son’s birth, as well they should, the Beracha is nevertheless delayed until the Berit Mila.
Summary: Parents should not recite a Beracha at the time of a boy’s birth, and should instead have in mind that the Beracha of "She’heheyanu" which the father recites at the Berit should refer as well to the occasion of the birth.
See Ben Ish Chai, Parashat Re'eh, Halacha 8
