Question: What is the significance of saying “Baruch Hu U’Varuch Shemo” (“Blessed is He and blessed is His Name”) when hearing the Name of G-d?
Answer: In the Song of Ha’azinu (Deuteronomy 32:3) it says: “When I call out the name of the L-rd, ascribe greatness to our G-d.” Sefer Chareidim cites a Midrash that explains: “Moses said to Israel: when I mention the Name of the Holy One, blessed be He, you should ascribe greatness to our G-d”—from here comes the practice of responding “Baruch Hu U’Varuch Shemo” upon hearing the Name.
The Talmud further states that even when mentioning a righteous human being one should bless him, as it says (Proverbs 10:7), “The memory of a righteous person is for a blessing.” All the more so, “when I mention the Righteous One of the universe—you should offer a blessing.” The author of Sefer Chareidim and other great authorities viewed this as a Rabbinic positive commandment, but most halachic authorities—and the Shulchan Aruch—mention it only as a custom, which began with the Rosh (about seven hundred years ago) in the name of his father.
Adapted from Sichat HaShevua