Great question!

According to the Talmud, the souls of all converts were actually present at Mount Sinai when the torah was given (Talmud - Shavuot 39a). This is derived from an interesting verse in the Torah, as I will discuss.

On the last day of Moses' life, he gathered all the Jewish people together who had wandered with him in the desert. There, God spoke to the entire nation of Israel, and said, "Not with you alone do I seal this covenant, but with whoever is here, standing with us today before the Almighty our God, and with whoever is not here with us today." (Deuteronomy 29:13) 

This verse is rather perplexing. We can understand God saying, "I seal this covenant... with whoever is here," because we can see that the people who are there are those who were wandering in the desert with Moses. But who is the group that God refers to as "whoever is not here with us today?!" According to the Talmud, this refers to converts who would convert in the future. It is from this, that we derive that the soul of all coverts were also at Sinai (Talmud - Shavuot 39a).

A close look at that verse in the original Hebrew will also show something startling. If you look at the first part of the phrase, "but with whomever is here" you will see that the last letters of the first four words (in Hebrew) actually spell out the word "Yitro" which was the name of Moses' father-in-law. Yitro was a righteous convert, and the archetypal covert of the Five Books of Moses. The fact that his name can be found in the verse "with whomever IS here" also indicates that converts of future generations were also there that day.

The Talmud, continuing this idea that converts already have a Jewish soul inside of them, uses a very interesting phrase when discussing Jewish laws of potential converts. It is written, "a convert who comes to convert..." The phrase begs the question - why does it say "a CONVERT who comes to convert..."? Rather, it should say, "a GENTILE who comes to convert..."! The reason is because they already have a Jewish spark inside of them.

Rabbi Yoel Schwartz writes in "Jewish Conversion" (www.feldheim.com): One of the most well-known converts was the Polish nobleman, Abraham ben Abraham. He converted to Judaism in the 18th century, and was sentenced by the church to death. He died sanctifying God's Name. It is said that even before his conversion, unidentifiable feelings, which testified to the greatness of his spirit, would overwhelm him every Sabbath.... Abraham ben Abraham traced the roots of converts to the giving of the Torah. "Although the nations rejected the Torah," Abraham ben Abraham is quoted as saying, "Individual members of those nations sought to accept it. Only the refusals of their peers prevented them from realizing their aspirations. The souls of these individuals appear in every generation as converts."