Excerpted with permission from Becoming a Jew (Jonathan David Publishers, Inc.).
The sages say that one of the virtues of the Jews in their exile in Egypt was that they did not alter their names. That would have signaled an altered worldview, the adoption of a new lifestyle, a quick and efficient scrapping of the past.
A change of name for the convert, following that logic, signals the embracing of a new philosophy, a new identification, a purposeful, mindful statement of intent for the long future. “A convert is as a newborn child, k’tinok she’nolad.” A new person needs a new name. That is why the rabbis instituted that converts should choose Hebrew names for their new Jewish lives. 
Changing Given Name a Choice, Not a Requirement
Many rabbis hold that converts should not only add a Hebrew name but also modify the given name used in pre-conversion years. Other rabbis differ, pointing to Ruth, the most famous female convert to Judaism, who did not change her Moabite name at all. Still other rabbis hold that all converts should be named Abraham or Sarah, the very names they were given when they “converted” to the service to God. Tradition imputes to them the constant activity of converting men and women to the worship of God. But Jewish communities never followed that advice.
Some rather strange names for converts surface in the times of the Talmud. One is “Son of Hay Hay,” another is “Son of Bog Bog.” The theory offered is that these converts were in danger of reprisals for defecting to Judaism and, in order to hide their convert status, did not use their Hebrew names or their spiritual patronymic, “son of Abraham, our father.” Rather, they devised names that subtly conveyed their convert origins, as for example, “Son of Hay, Hay,” which indicates that he is the spiritual heir of the two people who had the Hebrew letter hay added to their names, Abraham and Sarah. “Son of Bog Bog” did the same, only more secretively–the numerical total of the Hebrew letters bet and gimel, pronounced bog, is five, as is hay.
The choice is ultimately the convert’s, and should be made with full knowledge of the scope of names available, not only in terms of pleasant-sounding words, but of their meanings.
Converts Adopt Abraham and Sarah as Spiritual Parents
What is not the choice of converts is the identity of the parents. In Jewish life, a person is formally called by his or her given name, and as the son or daughter of the parent. (Reference is generally made to the father–except in illness or in danger, when compassion needs to be elicited and the person is referred to as being the child of the mother.)
While the convert’s [given] name is the convert’s own choice, Judaism requires, in all formal documents, legal proceedings, and religious functions such as being called to the Torah, an identification of parentage.
As the convert is technically considered to be a newborn child, reference to the parent must be of the spiritual parentage adopted by entering into the Covenant of Abraham. There must be a formal designation of the conversion that is plainly evident. That is why the convert is called “ben Avraham Avinu,” “son of our Father, Abraham,” or “bat Sarah Imenu,” “daughter of our Mother, Sarah.” In a Jewish marriage contract or divorce, it is not sufficient to write “child of Abraham,” but of “Abraham, our Father,” and “Sarah, our Mother,” in order to avoid any possible duplicity that might lead some to believe that the father was actually Jewish and the person’s name simply Abraham. Sometimes the word ha’ger, “the convert,” is appended to the name.
This naming pattern was required only of the first generation of converts. All subsequent generations refer to their own father’s Jewish name, without the convert appellation. The convert title appended to the name should be borne as a badge of spiritual courage and accomplished idealism. But it should be noted that this title is required only on formal occasions and documents. It need not obtain in personal, familial, and social life.
The time of the naming ceremony was held by some to be the same as for a Jewish-born male–at the circumcision rite. However, as the convert is at this point still not fully converted–not having completed the immersion–and therefore not yet a Jew, the naming ceremony should preferably be delayed until immediately after the immersion. It is generally recited at that time for both male and female converts.
The prayer recited is as follows (for males substitute the correct pronoun):
“Our God and God of our Fathers:
Sustain this woman in the Almighty’s Torah and in His commandments and may her name in Israel be _____________, the daughter of Abraham, our Father. May she rejoice in the Torah, and exult in the commandments. Give thanks to God, for He is good and His kindness is to all eternity.
May ____________, the daughter of Abraham, our Father, grow to become great. So may she enter the Almighty’s Torah, with His commandments and good deeds.”
At the conclusion of the entire ceremony, some versions add this prayer:
“Our God and God of our Fathers:
Enable this convert to succeed. Spread Your kindness over her. As You influenced her to find shelter under Your wings and to join Your people, so may You implant love and awe for You in her heart. Open her heart to Your teachings.
Lead her in the way of Your mitzvot [commandments]. May she merit to conduct herself in accordance with Your own attributes and may she always win favor in Your eyes.”
You can discover options for your Hebrew name on Kveller‘s Jewish name bank.
Avraham
Pronounced: AHVR-rah-ham, Origin: Hebrew, Abraham in the Torah, considered the first Jew.
Torah
Pronunced: TORE-uh, Origin: Hebrew, the Five Books of Moses.
