The Jewish Publication Society commentary on Genesis doesn’t offer up much different from the previous comments, except that if Sarai is derived from a nearby language (not Hebrew) it means “queen.”
The language is one level of analysis and the context, here of the covenant with Abraham, is another. Sarah is viewed as a prophetess in Judaism. Christians learn in Revelation that God has a secret name for each of us, and here, it seems, that name is revealed to each of Abram and Sarai. So, they need to look at each other in God’s terms, not in the old way.
That suggests an elevation in status of Sarai and points to her role in the genesis of Israel. (I’m way past the commentary here.) So, stretching my neck out here, God is perhaps pointing out that the nations shall come about through Sarah rather than through Hagar, who sires Abraham’s first born. So, changing her name might be the part of the prophecy and covenant that will be fulfilled through her.
A feminist view would say that Sarah goes a long way, in this respect, towards restoring the status of women which was reduced by the sin of Eve.
Symbolism? Adam and Eve start off the human race earlier in Genesis. Noah and his wife represent another new start to humanity after the flood. And a new expanded spiritual dimension begins with Abraham and Sarah. These represent progressions from family to tribe to nation, or are at least along that progression.
Isn’t Abraham referred to as “our father in faith?”
More shaky ground: Isn’t Abraham mentioned more in the New Testament than the other patriarchs? (74 times by my count in Strong’s Concordance.)