“ויאיר בן מנשה הלך וילכד את חותיהם ויקרא אתהן חות יאיר. ונבח הלך וילכד את קנת ואת בנתיה ויקרא לה נבח בשמו” (ל”ב,מ”א,מ”ב)
In the sefer אורחות רבינו the Steipler zt”l says in the name of the Chazon Ish, that naming a child two names after two different people is a completely new name. Therefore the child is not considered named after any of them. It is only considered a remembrance after them. The Chazon Ish brought a proof to this from the end of Parshas Matos. It says that Yair captured חותיהם and he called it חות יאיר. The next passuk says that נבח captured קנת ואת בנותיה and he called it נבח בשמו – he named it after his own name. By יאיר it does not say he called it בשמו like it says by נבח. Says the Chazon Ish because by יאיר he added a name חות therefore it is not his name. However נבח named his city after his name without adding on another name, that is why it is בשמו. We learn from here that adding on a name changes the name entirely and makes it a new name.
When the first son of Reb Isaac Ausband zt”l was born he wanted to name him after his father-in-law Reb Avrohom Yitzchok Bloch hy”d. Reb Ausband’s name however was Yitzchok Isaac, so he asked Reb Eliezer Silver zt”l what name he should call his son. Is Avrohom Yitzchok one name and therefore it is not the name Yitzchok or are they considered two separate names for one person and he would only be able to call him Avrohom? Reb Silver sent him a short telegram stating that he can call his son Avrohom Yitzchok without a doubt, ending off that he has proofs for this ruling from the Torah. Reb Isaac was very surprised, where in the Torah does it discuss naming after another person? When they met some time later Reb Silver told him the same proof as the Chazon Ish.
Reb Dovid Zucker Shlit’a as well as Reb Silver ask on this ruling from Parshas Devorim. The passuk there says יאיר בן מנשה לקח את כל חבל…ויקרא אותם על שמו את הבשן חות יאיר עד היום הזה. We see that חות יאיר was also called על שמו. They both answer that there is a difference between על שמו and לשמו. When the torah says על שמו, that means as a remembrance. לשמו means his actual name.
Reb Chaim Kanievsky shlit”a in his sefer טעמא דקרא brings from the Chazon Ish that if a child is being named after someone with two names, both names should be given. By giving only one name it is an entirely different name. It is still considered a remembrance though, if only one name is given.
Reb Moshe Shternbuch shlit”a as well as others are of the opinion that it is better not to name a child after two different people. It could have an improper effect on him. However the poskim say that if the husband and wife can’t agree if the child should be named after his father or her father then he should be called after both.
Prepared by R’ Avrohom Yehoshua Ziskind
 Sources:עטרת דודאים,מסורת הברית