Here's an excerpt from http://www.claudemariottini.com/user/Roberts%20Beula%20Land%20Complete.pdf:

"The significance of a new name is also noted by Freehof who states, "A new name is a symbol of a new and better life." In Israelite culture, the giving of a new name also was "a sign that the receiver of the name is coming under the authority of the giver of the name." In giving Zion a new name, God was calling Zion to be under his authority.

The word "Hephzibah" is translated from Hebrew as "My Delight is in Her". This word is expressing God's love and tenderness towards His people. According to Haak, the word "Hephzibah" is "found in only two places in the Bible. First, it was what the God-fearing
Hezekiah called his wife. In II Kings 21:1 we learn that Hezekiah's wife was called "Hephzibah" and Hezekiah could say concerning his wife, 'My delight is in her'." God is delighting in his people. In the last part of Isaiah 62: 4 God is calling them into a relationship "where thy land shall be married." The marriage imagery is used to indicate that Yahweh is inviting his people to be in relationship with Him. This is what the word Beulah is signifying. Yahweh is the groom, Israel the bride. The marriage theme is used throughout the Old Testament and the author of verse 4 uses it to "describe the relationship between Yahweh and his people, much as his predecessors, Hosea, Jeremiah, Ezekiel and Second Isaiah, had done."35 Isaiah 62:4-5 gives a complete picture of the marriage imagery"...