Leithart observes that the three uses of the word on the pillar that are in the temple (Rev. 3:12) make reference to the Triune Name. He writes:
Jesus promises to write a triple name on the pillars that are in the temple (Revelation 3:12). The three uses of the word ????? are the name of “My God,” the name of the city, which is New Jerusalem, and Jesus’ own new name. This has got to be a Trinitarian formula. “My God” is definitely the Father, and Jesus’ own Name is the Name of the Son. The sticking point is that the third Name is not the name of the Spirit but of the city. But there are reasons to think that the Name of the city is the Name of the Spirit, that New Jerusalem is a way of speaking about the presence of the Spirit.
At the end of the book of Revelation, the Spirit and the Bride speak with a united voice, calling on the Bridegroom to come (22:17). The seven Spirits of God have formed and inhabited a bride, so that the Bride speaks with the Spirit’s voice. The description of the Bride/New Jerusalem descending from heaven reminds us of the dove descending from heaven onto Jesus at His baptism (Matthew 3:16; Mark 1:10; Luke 3:22; John 1:32-33). In Revelation 3:12, the phrase “descending from heaven” is nearly identical to the phrase in John 1:32 . That the Spirit is pictured as a dove points to an association with the Bride, since the Bride is the “dove” of the Song of Songs (2:14; 5:2). By receiving the Spirit as a dove coming from heaven, Jesus is joined to the Bride.
Thus, it makes sense that the triple name of God would put the city where we would expect the Spirit. And that means that the city, the bride from heaven, is incorporated into the divine name. The Bride becomes part of the family and bears the family name.