In the central and southern areas of the state, free legal help and advice for single parents is offered through the Land of Lincoln Legal Assistance Foundation. They have several offices that specialize in helping single mothers cope with difficult family situations. To find the office that serves the county where you live, look at the list here.
The 36 counties of northern Illinois are served by Prairie State Legal Services. This nonprofit organization helps low-income parents find the help they need in handling legal problems related to taxes, unemployment insurance, health care, housing and evictions, domestic abuse, and family benefits programs. You can find their office telephone numbers and locations here. They provide counseling but not legal help in family law cases like child support, custody, and divorce, but your counselor will be able to direct you to the groups in your town that provide free or low-cost legal aid for those areas.
If you live in Chicago then you can get help from the Legal Assistance Foundation. Call their Client Screening Unit or apply for help on their website here.