What was the name of this lady slave and how did she help the father of David Hamelekh? Thank you

ב"ה Shalom In your question, I assume you are referring to the midrash about Yishai, King David's father who decided to have children from his maidservant. There is no mention of her name in this midrash. Even the name of Yishai's wife is not mentioned in the Tanach, but it is mentioned in the Talmud . (Baba Batra 91a), Nitzevet bat Adel. After Yishai had children with his wife, he had doubts again that possibly the prohibition mentioned in the Torah (Devarim 23: 4-5) that one may not marry an Amonite or a Moabite applies to women as well. The Talmud says that the prohibition applies to men only but not the women from Amon. However, in that generation this halacha had not fully entered everyone's recognition. Therefore he deliberated with himself, that since he is a descendant of Ruth who was a Moabite, then he himself is a Moabite and he shouldn't be married to his own wife who is a full fledged Jewess. He therefore made a condition in regard to his maidservant. He told her that she is free from her enslavement under the following condition. If the Torah included a Moabite woman in the prohibition of marriage, then she is NOT free and he will marry her since he is not really Jewish and he may marry a non-Jewish maidservant. However, if the prohibition is only in regard to Moabite men then she is free and now he may marry her as well. The midrash goes on to tell the righteousness of Yishai's wife and how she dealt with this. All the best