The Shulhan Aruch (Yoreh De’a 344:4) rules that if an infant dies, Heaven forbid, within thirty days after birth, or in the case of a stillborn, Kaddish is not recited and mourning not observed. By the same token, no Hashkaba (memorial prayer) is recited for the deceased infant in these cases. An infant who does not survive for thirty days ascends directly to Gan Eden and does not require any observances. Therefore, there is no need for Kaddish, mourning or Hashkaba for a stillborn or an infant who dies within the first thirty days after birth.
The Hid"a (Rav Haim Yosef David Azulai, 1724-1806) writes that if a couple has several children that do not survive for thirty days, they should name their next boy Bension, as this name is a Segula ("charm") for a child’s survival.
There is a custom to add a second name to a seriously ill patient. For example, if a patient’s name is Abraham, his name might be changed to Refael Abraham. If the patient recovers from his illness after the new name is added, then the new double name is what should be used, even after his death. Thus, after his passing, both names should be engraved upon the Maseba (gravestone) and should be mentioned in the Hashkaba. Similarly, if people recite Tehillim in his memory, they should have in mind the new name. However, if the name addition was not effective in curing the patient, and he died from that illness, then the new name is not used. In the example mentioned above, then, the deceased would be referred to only as Abraham, and not as Refael Abraham.
In his work Haim Sha’al (vol. 1, 71:6), the Hid"a writes that when reciting the Hashkaba, one should recite only the text found in the Siddurim, and not add any praises or honorifics to describe the deceased. Our Sages have taught us that oftentimes "Kol Ha’mosif Gore’a" – adding extra material actually detracts, and undermines the desired goal. And thus in the case of the Hashkaba, it is preferable to adhere to the standardized text, and not attempt to embellish it by adding praises.
Summary: If, God forbid, and infant does not survive the birth or dies within thirty days of birth, there is no Kaddish, mourning, or Hashkaba prayer. If a name was added to the name of an ill patient, but the patient died from that illness, the new name is no longer used in reference to the deceased. If, however, the patient recovered, the second name should be used, even after his death. When reciting the Hashkaba prayer, one should adhere to the standardized text and not add praises or honorifics for the deceased.
