Especially 6:27 -- 27 And they shall put my name upon the children of Israel, and I will bless them.
Numbers 6:23 - 27
ESV - 23 Speak to Aaron and his sons, saying, Thus you shall bless the people of Israel: you shall say to them. 24 The Lord bless you and keep you.
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As I read the passage referenced in the question (specifically, Numbers 6:27), the priests (Aaron and his sons/descendants) put God's name on the Israelites by speaking the blessing contained in Numbers 6:24-26, in which the priests invoked the name of God (that is, the LORD (Exodus 3:13-15)) in blessing and keeping the people, in making His face to shine upon the people [that is, to be favorably disposed to them, in contrast to turning His face away from them] and to be gracious to them, and that He look upon the people with favor and give them peace. These blessings came to Israel specifically from God and in His name, and identified Israel as His people or possession, in metaphorically the same way that people identify earthly items as theirs by putting their names on them.
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