Generally, a grandparent has no legal or inherent parental (even visitation) rights, as pertains to grandchildren.
There are SOME exceptions, and Illinois does offer a slight remedy, something akin to permissive rights, not a right, more of a privilege.
Illinois law refers to grandparent visitation as a "privilege." It's not a right. You do have the right to go to court and ask for visitation, but a court is not required to give you any. Under Illinois law you must show that denying visitation will harm the child mentally or physically, and that you are being unreasonably denied visitation by a parent. The law also lets great-grandparents and siblings ask for visitation.
Before going to court, talk with the child's mother. Try to work out some visitation by agreement. If that doesn't work, your next step would be to file a petition for grandparent visitation. You'd file that petition as a new proceeding in your son's divorce case. In other situations, where there's no divorce or custody case, grandparents can file their own case if the grandchild's parents are separated, or if a parent has died.
Once that petition is filed, you can still try to work out an agreement for a visitation order. Otherwise, you'll have to have a trial, and present witnesses and evidence.
If the mother is considered a fit parent, then the court must follow her wishes unless there is a danger of harm to the child. Judges will usually try to maintain a healthy and established relationship between grandparents and any grandchild(ren).
On the other hand, grandparent visitation will be denied or permanently revoked if it's being used to get around a restriction on a parent's visitation rights.
That means that if your son or daughter has had their visitation restricted in some way, you'll have to prove that your visitation won't violate those restrictions.
A couple final notes:
> Grandparents normally aren't granted visitation time that equals parental visitation time.
> Lawyers in Illinois will probably charge about $500, or much more, to handle a grandparent visitation case.