Generally speaking, the statutes appear to allow this provided it has been more than 3 years since the dissolution - see section 12.43 of the Business Corporation Act. I generally recommend using a new name so as to avoid confusion and potential liabilities - you may question how easy it will be to explain to the IRS that you need a new EIN for a company with the same name etc. Also, you should consider whether another entity such as an LLC makes more sense than an s-corp for your situation. This is all just general discussion and should not be considered legal advice for your specific situation for which it is recommended you retain an attorney.
I don't believe that the Illinois Secretary of State would even permit the registration of a company by the same exact name. It appears that you have two options: 1) pay all of the back fees, etc., as suggested by the Secretary of State's office, if you wish to use that same exact name; or 2) start a new company and pick any other name that isn't already being used or otherwise protected.
Best to just create a brand new entity with a new name to avoid any mistake or potential creditors hunting your new entity down if it had the same name as the old (but dissolved) entity. I had a client get sued on such a circumstance and even thou they had no liability whatsoever their issuance company decided it was cheaper to throw some dollars towards settlement than to fight in court to prove the former entity was nit them ......is common sense for sure.
My answer is not intended to be giving legal advice and this topic can be a complex area where the advice of a licensed attorney in your State should be obtained. Please click "helpful" or "best answer" if my answer added any value or add a "comment" if you have more info for me to help you get a better answer.