Naming one’s child after one’s parent who has passed away:[1]

It is customary for children to name their own children after the names of their parents.[2] Doing so is included in the Mitzvah of honoring one’s parents.[3] [This should be done even if the parent passed away at a young age.[4] If the father passed away prior to the birth of the son, then the custom is to name the child after his deceased father.[5]]

Grandparent: Naming a child after one’s grandparent is likewise considered an honor to one’s parents.[6] [Nevertheless, naming one’s child after one’s own parent comes before naming them after one’s grandparent.[7]]

Adding a name:[8] It is permitted for one to add another name to the original name of the parent and call the child by both names or by only the parents name. However, he should not call the child by only the second name which is not the name of the parent.

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[1] Pesakim Uteshuvos 240:33; Nitei Gavriel 72:1

[2] Bereishis Raba 37:10; Torah Temima Bereishis 32:11; Otzer Habris 5:4 footnote 2; Shevach Habris 20:6

The reason: Several reasons are mentioned behind this custom, including: 1) To document our lineage which would otherwise be difficult to document in times of exile. [Bereishis Raba ibid] 2) During times of exile is important that we remember the names of our ancestors. [Torah Temima ibid] 3) It is done as a Segula for the child to inherit the good traits of the ancestors. 4) It is done in order to commemorate the name of the ancestor and give him satisfaction above which is a benefit for his soul. [Otzer Habris ibid]

[3] Mili Dechassidusa on Sefer Chassidim 477; Sheilas Shalom Tinyana 243; Shem Mishimon Y.D. 22; Sdei Chemed Mareches Chaf Kelal 24; Maaneh of Rebbe in Shavuon Kefar Chabad Gilyon 954

[4] Maaneh of Rebbe in Shavuon Kefar Chabad Gilyon 954

[5] Nitei Gavriel 72:6 footnote 8

[6] Betzel Hachochmah 3:108

[7] Rivam Shteinbach 58; Nitei Gavriel 72:2

[8] Beir Moshe 1:60