Often the best way to find a lawyer of any type is to get personal referrals. Talk to people you know that have been through the same situation you have. You can meet people at support groups and they might be able to refer you to a great birth injury lawyer. A birth injury attorney will have experience dealing with these types of cases and will be able to help you with the court process.
If you don't have any people available around you to refer you to a good lawyer, you might be able to find one that you feel comfortable with online. Often a birth injury attorney will have set up a Web site with information about his/her practice and beliefs, and this can make you feel more at ease when you set up an appointment with them. Birth injury attorneys with websites online also give you a chance to "preview" the type of law that they practice, that way you can get to know the lawyers before choosing one.