🔼The name Sarai in the Bible
Sarai is the original name of Sarah, the wife and half-sister of Abraham (same father, namely Terah: Genesis 20:12). Before the conception of Isaac, Sarah was called Sarai. Possibly, the name-change indicates a step from local to global, or specific to general.
Read about the Hebrews' powerful invention of the vowel notation — which made the writing of names such as Sarah and Sarai possible — in our article on Yahweh, the Name of the Lord.
🔼Etymology of the name Sarai
The name Sarai comes from the same word as the name Sarah, which means princess:
The final letter י (yod) of the name Sarai gives the noun a possessive form, and would mean My Princess.
For a meaning of the name Sarai, NOBSE Study Bible Name List groups it without further comment with Sarah and seems to translate both names with Princess. Jones' Dictionary of Old Testament Proper Names discusses the name Sarai under the article on Sarah, but translates Sarai with My Princess. BDB Theological Dictionary doesn't translate but does connect Sarai and Sarah to the noun שׂרה meaning princess.