Welcome to Roots Chat. I am an adoptee, so I have some experience in your sort of search.
There are death records/indexes for Cook County, Illinois on
www.familysearch.org, Ancestry.com, and
www.cookcountygenealogy.com. I don't see any deaths for a Fiona Lopez from that time frame, and the only Fionas with different last names that I see are either too young or too old to have given birth around then.
If I were you, I would look into your father's previous marriage. If you have a name of his first wife, you can go from there and see if she's deceased or what happened to her. If the marriage was in Illinois, there are some marriage records in the sites I mentioned above.
I think the suggestion above about getting your original birth certificate is a good one, so at least you would know one way or the other if the one you have is the original.
I don't know if they currently ship to Mexico, but autosomal DNA testing is another thing to consider. If your biological mother has any relatives who have tested, you may be able to figure out who she was. I used Ancestry.com for DNA testing, and it was a success for me.
www.dnaadoption.com and
http://www.dna-testing-adviser.com/ have lots of helpful information about the topic.
One other thing I would recommend - keep in mind that people's memories can be faulty and, sadly, sometimes people will mislead you. There is usually a grain of truth in family stories, but the actual truth can be a bit different. I wish you luck with your search.