Illinois: We have a renter that is 3 months behind and we are giving her a 5 day notice today. We sent her and her co-signer (who does not live with her) a notice in early Dec with all the rent and late fees and damage costs stating that we would be filing against both if rent was not paid by Jan 1st.
Do we have to give the co-signer a 5 day notice separately or can we put her name on the notice for our tenant.

You can use one notice, but service has to be done separately.

If you are a new landlord it is always best to have a professional legal person show you the proper forms for your location and even specifics your local courts may normally ask for.

I often say that a landlord should know and have prepared eviction notices up the day before they buy a rental property.

Normally it is one notice and both are served.