What is Ruth's full Jewish name?

Ruth's name when she born is Ruchel Dwarjra Zylska. This is provided to the reader on the first page of the book.  On the second page, we are told her name was changed to Rachel Deborah Shilsky.

Ruth was born in Poland on April Fool's Day, 1921 (1).  This means that when her family came to America, her name, probably along with the names of the rest of the family, was changed to be more "American."  You can see the similarity between her birth name and her American name, a similarity that was often preserved, keeping some of the old country, but changing it enough to assimilate in an English-speaking country.

People who are Jewish often have two names, a Hebrew name and a "secular" one.  In many Jewish families, the tradition is to name a child after someone who is deceased, in honor of that person, of course, but also because there is some superstition attached to naming a child after someone who is still living.  Frequently, a child will have two Hebrew names, one from someone deceased on both sides of the family.  That is so neither side of the family will feel left out!  Hebrew names are used for ceremonial purposes, weddings, bar and bat mitzvahs, confirmations, etc.  They are also used as family nicknames.

But in the "old country," where Ruth was born, the Hebrew name and the everyday name were sometimes the same, with no distinction for religious or ceremonial purposes.  This is still true in some Orthodox and Chasidic Jewish families.

In Ruth's case, her original name, "Ruchel" was the Hebrew form of Rachel, and "Deborah" is the equivalent of the Hebrew name "Devorah."  Interestingly, she still chose a Biblical name when she made her final change, but one that is from a story about a woman who was not Jewish at all.