I've heard this argument:
Adam and Eve are imperfect types of Jesus and Mary, the new Adam and Eve, respectively. Because Old Testament types can never be superior to their New Testament fulfillment, and because Eve was born sinless, Mary must have been born sinless as well.
I don't think it holds up to basic scrutiny, though. Eve was born sinless because no sin was in the world when she was created. Similarly, the argument opens up all sorts of absurd counterpoints. Adam was the first man to ever be created, but Jesus Christ was something like the 1,998,973 man to be created, etc.