There are several different ways you could go at this and all of them have a very slim chance of resolving this family mystery.
First of all, Y-DNA (surname DNA) testing would not be relevant for you, since woman do not have the Y-chromosome. This particular chromosome is handed down from father to son, so it can be used to help verify paper trails of paternal (father's father's father) ancestry. Only males can be tested, but females can use this type of testing as well if they can use the DNA of certain male relatives.
There is another test that would be relevant for you, called Mitochondrial DNA testing (mtDNA). Mitochondrial DNA is passed down from mother to child, so it can be used to trace maternal ancestry (mother's mother's mother). Normally, mtDNA would be perfect for resolving something like this, but since everybody in your mystery is so closely related, you probably won't be able to get enough testing resolution to find out.
If you want to use mtDNA, you need to take a test, and you also need to find a child of the other woman to get tested to compare your results to. Both tests would need to be of the highest resolution to increase your chances of success. If the two tests DID match, you would be unable to resolve the mystery (95+ percent chance). If the two tests did NOT match entirely, you would have a 1/3 chance of proving your theory. Basically, you would just keep testing relatives (assuming enough are available) until you could track down in which generation the mutation occurred. Since each test costs $495, this would be an expensive process, with only a slim chance of success. However, even in the event of failure, you would still learn a great deal about your "deep" ancestry (where your ancestors lived thousands of years ago) and the test results would be useful for other genealogical research.Order the Full Sequence test for $495 by clicking here
You could also look into autosomal DNA tests. Basically, these are a form of paternity testing (or maternity testing in this case) that tests all the chromosomes, not just the Y or mtDNA. Since autosomal DNA tests are rarely used in genealogy, I know little about it... A representative of one of the companies that offers this type of testing might be able to tell you more. These types of tests are not normally used to test ancestry so many generations back, so I doubtful that they would be of much use to you.
I'm sorry that I'm not able to provide a clearer or concise answer, but your question deals with some pretty tricky issues that are difficult to explain. I hope I gave you enough information to be useful. If you have other questions or wish to discuss this further, feel free to respond.