We love her so much, but she has completely lost her memory at this point. All she can do is sing lullaby songs that she knew in her youth :(
Her children (my mother, aunt, and uncle) are not sure where she was originally born. They have narrowed it down to two possible locations in New Jersey but they are still not totally sure which of the two was her actual birthplace. We would ask her, but she has been mentally gone for years :(
I have her full Maiden name, including her middle name. I also have the first, last and middle name of her mom and dad. What is the best way to figure out where she is from? I would obviously prefer not to pay for any services, but I would probably be willing to if necessary.
Depending on her age, you might be able to find her (or her parents) in old census records. Historical census records are maintained by the National Archives, and you can learn more about those records here: https://www.archives.gov/research/genealogy/census/about
The most recent census records that are publicly available are from the 1940 census. There are a number of paid services, but you may be able to find what you need for free. Your local public library may offer online access to one or more services that can help you find what you're looking for, and some websites offer free search with account creation.
If you have any information about the family's old address, that can help narrow the search considerably (though I expect you don't have that, given the situation).
There is a question on the census for "birthplace", but this usually lists a state or country rather than a specific town. You may be able to figure out the town where she was born based on where the family was living at the time of the census (or at least narrow it down based on what you already know).
If the 1940 census is too recent, you'll need to wait for the 1950 census. Census records become publicly available after 72 years, so the 1950 records won't be available until 2022.
Alternatively, you might be able to request a copy of her birth record from the State of New Jersey (if you're sure that's where she was born). The site that I found seems to say you can only request records as "genealogical records" if the person is deceased and born more than 80 years ago, so this option may not work for you right now: https://www.state.nj.us/health/vital/order-vital/genealogical-records/
If you can't find what you need in the census records, you might consider contacting someone who handles vital records for the State of New Jersey and see if there are any options or exceptions for someone in your situation.
Good luck. I hope you find what you're looking for.