Are you wanting to specifically be a "civil" public defender? I ask, because the majority of public defenders do criminal law. Many leave after a few years and go into private practice, which can be quite lucrative.
There are some civil public defender positions. The ones I have heard of are in family law and education law. Let's say the state child protective services is trying to take away custody of a child, and the parents cannot afford an attorney. I believe one may be appointed for the parents, and, a separate one for the child. Or suppose a school is not giving a special needs child the proper education. I believe there can be lawyers appointed in those cases. Fair housing also might have court appointed attorneys. Suppose a person needs an apartment modified for a wheelchair and is having a fight with the landlord. I am not sure of other areas.
As you can see, many of these issues revolve around Federal Law and Civil Rights. Law school will give you some choices on the courses you take, so you might want to consider some that deal with these areas. You should also get the opportunity to do some law clinic work your 2nd or 3rd year, which will give you some hands-on experience.
I apologize for not knowing more. I did not want you to not have an answer to your question!